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# Tutorials, by Mathematical Problem

TODO: Add link to Python example here

## Linear elliptic PDE on a 2D grid

WHAT THIS EXAMPLE DEMONSTRATES:

- Using command line options

- Using Linear Solvers

- Handling a simple structured grid

FURTHER DETAILS:

- Mathematical description of the problem

- the source code

DO THE FOLLOWING:

- Compile `src/ksp/ksp/tutorials/ex50.c`

      $ cd petsc/src/ksp/ksp/tutorials
      $ make ex50

- Run a 1 processor example with a 3x3 mesh and view the matrix assembled

      $ mpiexec -n 1 ./ex50  -da_grid_x 4 -da_grid_y 4 -mat_view

  Expected output:

      Mat Object: 1 MPI process
        type: seqaij
      row 0: (0, 0.)  (1, 0.)  (4, 0.)
      row 1: (0, 0.)  (1, 0.)  (2, 0.)  (5, 0.)
      row 2: (1, 0.)  (2, 0.)  (3, 0.)  (6, 0.)
      row 3: (2, 0.)  (3, 0.)  (7, 0.)
      row 4: (0, 0.)  (4, 0.)  (5, 0.)  (8, 0.)
      row 5: (1, 0.)  (4, 0.)  (5, 0.)  (6, 0.)  (9, 0.)
      row 6: (2, 0.)  (5, 0.)  (6, 0.)  (7, 0.)  (10, 0.)
      row 7: (3, 0.)  (6, 0.)  (7, 0.)  (11, 0.)
      row 8: (4, 0.)  (8, 0.)  (9, 0.)  (12, 0.)
      row 9: (5, 0.)  (8, 0.)  (9, 0.)  (10, 0.)  (13, 0.)
      row 10: (6, 0.)  (9, 0.)  (10, 0.)  (11, 0.)  (14, 0.)
      row 11: (7, 0.)  (10, 0.)  (11, 0.)  (15, 0.)
      row 12: (8, 0.)  (12, 0.)  (13, 0.)
      row 13: (9, 0.)  (12, 0.)  (13, 0.)  (14, 0.)
      row 14: (10, 0.)  (13, 0.)  (14, 0.)  (15, 0.)
      row 15: (11, 0.)  (14, 0.)  (15, 0.)
      Mat Object: 1 MPI process
        type: seqaij
      row 0: (0, 2.)  (1, -1.)  (4, -1.)
      row 1: (0, -1.)  (1, 3.)  (2, -1.)  (5, -1.)
      row 2: (1, -1.)  (2, 3.)  (3, -1.)  (6, -1.)
      row 3: (2, -1.)  (3, 2.)  (7, -1.)
      row 4: (0, -1.)  (4, 3.)  (5, -1.)  (8, -1.)
      row 5: (1, -1.)  (4, -1.)  (5, 4.)  (6, -1.)  (9, -1.)
      row 6: (2, -1.)  (5, -1.)  (6, 4.)  (7, -1.)  (10, -1.)
      row 7: (3, -1.)  (6, -1.)  (7, 3.)  (11, -1.)
      row 8: (4, -1.)  (8, 3.)  (9, -1.)  (12, -1.)
      row 9: (5, -1.)  (8, -1.)  (9, 4.)  (10, -1.)  (13, -1.)
      row 10: (6, -1.)  (9, -1.)  (10, 4.)  (11, -1.)  (14, -1.)
      row 11: (7, -1.)  (10, -1.)  (11, 3.)  (15, -1.)
      row 12: (8, -1.)  (12, 2.)  (13, -1.)
      row 13: (9, -1.)  (12, -1.)  (13, 3.)  (14, -1.)
      row 14: (10, -1.)  (13, -1.)  (14, 3.)  (15, -1.)
      row 15: (11, -1.)  (14, -1.)  (15, 2.)

- Run with a 120x120 mesh on 4 processors using superlu_dist and view the solver options used

      $ mpiexec -n 4 ./ex50  -da_grid_x 120 -da_grid_y 120 -pc_type lu -pc_factor_mat_solver_type superlu_dist -ksp_monitor -ksp_view

  Expected output:

        0 KSP Residual norm 3.039809126331e+00
        1 KSP Residual norm 2.395703277441e-14
      KSP Object: 4 MPI processes
        type: gmres
          restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
          happy breakdown tolerance 1e-30
        maximum iterations=10000, initial guess is zero
        tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
        left preconditioning
        using PRECONDITIONED norm type for convergence test
      PC Object: 4 MPI processes
        type: lu
          out-of-place factorization
          tolerance for zero pivot 2.22045e-14
          matrix ordering: external
          factor fill ratio given 0., needed 0.
            Factored matrix follows:
              Mat Object: 4 MPI processes
                type: superlu_dist
                rows=14400, cols=14400
                package used to perform factorization: superlu_dist
                total: nonzeros=0, allocated nonzeros=0
                  SuperLU_DIST run parameters:
                    Process grid nprow 2 x npcol 2
                    Equilibrate matrix TRUE
                    Replace tiny pivots FALSE
                    Use iterative refinement FALSE
                    Processors in row 2 col partition 2
                    Row permutation LargeDiag_MC64
                    Column permutation METIS_AT_PLUS_A
                    Parallel symbolic factorization FALSE
                    Repeated factorization SamePattern
        linear system matrix = precond matrix:
        Mat Object: 4 MPI processes
          type: mpiaij
          rows=14400, cols=14400
          total: nonzeros=71520, allocated nonzeros=71520
          total number of mallocs used during MatSetValues calls=0
            has attached null space

- Run with a 1025x1025 grid using multigrid solver on 4 processors with 9 multigrid levels

      $ mpiexec -n 4 ./ex50 -da_grid_x 1025 -da_grid_y 1025 -pc_type mg -pc_mg_levels 9 -ksp_monitor

  Expected output:

        0 KSP Residual norm 2.152827139129e+01
        1 KSP Residual norm 3.672855969144e-01
        2 KSP Residual norm 1.575285029053e-02
        3 KSP Residual norm 3.693894201785e-04
        4 KSP Residual norm 8.776528268726e-06

## Nonlinear ODE arising from a time-dependent one-dimensional PDE

WHAT THIS EXAMPLE DEMONSTRATES:

- Using command line options

- Handling a simple structured grid

- Using the ODE integrator

- Using call-back functions

FURTHER DETAILS:

- Mathematical description of the problem

- the source code

DO THE FOLLOWING:

- Compile `src/ts/tutorials/ex2.c`

      $ cd petsc/src/ts/tutorials
      $ make ex2

- Run a 1 processor example on the default grid with all the default solver options

      $ mpiexec -n 1 ./ex2 -ts_max_steps 10 -ts_monitor

  Expected output:

      0 TS dt 0.00847458 time 0.
      1 TS dt 0.00847458 time 0.00847458
      2 TS dt 0.00847458 time 0.0169492
      3 TS dt 0.00847458 time 0.0254237
      4 TS dt 0.00847458 time 0.0338983
      5 TS dt 0.00847458 time 0.0423729
      6 TS dt 0.00847458 time 0.0508475
      7 TS dt 0.00847458 time 0.059322
      8 TS dt 0.00847458 time 0.0677966
      9 TS dt 0.00847458 time 0.0762712
      10 TS dt 0.00847458 time 0.0847458

- Run with the same options on 4 processors plus monitor convergence of the nonlinear and linear solvers

      $ mpiexec -n 4 ./ex2 -ts_max_steps 10 -ts_monitor -snes_monitor -ksp_monitor

  Expected output:

      0 TS dt 0.00847458 time 0.
          0 SNES Function norm 1.044373877325e+01
            0 KSP Residual norm 6.321628301999e-02
            1 KSP Residual norm 6.526906698930e-03
            2 KSP Residual norm 3.007791706552e-03
            3 KSP Residual norm 4.405713554522e-04
            4 KSP Residual norm 1.328208424515e-04
            5 KSP Residual norm 1.644811353135e-05
            6 KSP Residual norm 2.166566242194e-06
            7 KSP Residual norm 1.586470130595e-16
          1 SNES Function norm 7.337449917266e-04
            0 KSP Residual norm 3.134728198020e-06
            1 KSP Residual norm 3.906901802932e-07
            2 KSP Residual norm 9.017310449502e-08
            3 KSP Residual norm 3.016048824996e-08
            4 KSP Residual norm 5.545058115802e-09
            5 KSP Residual norm 1.173289407756e-09
            6 KSP Residual norm 2.688949492463e-10
            7 KSP Residual norm 6.425931432155e-21
          2 SNES Function norm 5.661678516636e-11
      1 TS dt 0.00847458 time 0.00847458
          0 SNES Function norm 1.044514631097e+01
            0 KSP Residual norm 6.331279355759e-02
            1 KSP Residual norm 6.535292747665e-03
            2 KSP Residual norm 2.987562112557e-03
            3 KSP Residual norm 4.407960732509e-04
            4 KSP Residual norm 1.310867990310e-04
            5 KSP Residual norm 1.643512035814e-05
            6 KSP Residual norm 2.157619515591e-06
            7 KSP Residual norm 9.439033917013e-17
          1 SNES Function norm 7.215537347936e-04
            0 KSP Residual norm 3.094217119938e-06
            1 KSP Residual norm 3.853396165643e-07
            2 KSP Residual norm 8.869933627178e-08
            3 KSP Residual norm 2.968861004222e-08
            4 KSP Residual norm 5.442261222566e-09
            5 KSP Residual norm 1.152789140987e-09
            6 KSP Residual norm 2.628606842614e-10
            7 KSP Residual norm 3.895271394296e-21
          2 SNES Function norm 5.533643492917e-11
      2 TS dt 0.00847458 time 0.0169492
          0 SNES Function norm 1.044653068944e+01
            0 KSP Residual norm 6.340852763106e-02
            1 KSP Residual norm 6.543595844294e-03
            2 KSP Residual norm 2.967488018227e-03
            3 KSP Residual norm 4.410154015305e-04
            4 KSP Residual norm 1.293935090965e-04
            5 KSP Residual norm 1.642264605661e-05
            6 KSP Residual norm 2.149306177473e-06
            7 KSP Residual norm 1.303479711243e-16
          1 SNES Function norm 7.096638555936e-04
            0 KSP Residual norm 3.054510440042e-06
            1 KSP Residual norm 3.801000498387e-07
            2 KSP Residual norm 8.725839831972e-08
            3 KSP Residual norm 2.922750507411e-08
            4 KSP Residual norm 5.341779435787e-09
            5 KSP Residual norm 1.132841361166e-09
            6 KSP Residual norm 2.569880731718e-10
            7 KSP Residual norm 5.983404585705e-21
          2 SNES Function norm 5.236235205743e-11
      3 TS dt 0.00847458 time 0.0254237
          0 SNES Function norm 1.044789247558e+01
            0 KSP Residual norm 6.350349403220e-02
            1 KSP Residual norm 6.551817537718e-03
            2 KSP Residual norm 2.947571742346e-03
            3 KSP Residual norm 4.412293238928e-04
            4 KSP Residual norm 1.277401751314e-04
            5 KSP Residual norm 1.641065933559e-05
            6 KSP Residual norm 2.141569821046e-06
            7 KSP Residual norm 7.468211737013e-17
          1 SNES Function norm 6.980655055866e-04
            0 KSP Residual norm 3.015586472511e-06
            1 KSP Residual norm 3.749683834693e-07
            2 KSP Residual norm 8.584934528817e-08
            3 KSP Residual norm 2.877684460640e-08
            4 KSP Residual norm 5.243554447127e-09
            5 KSP Residual norm 1.113425160917e-09
            6 KSP Residual norm 2.512722872950e-10
            7 KSP Residual norm 3.506711593308e-21
          2 SNES Function norm 5.113451407763e-11
      4 TS dt 0.00847458 time 0.0338983
          0 SNES Function norm 1.044923221792e+01
            0 KSP Residual norm 6.359770143981e-02
            1 KSP Residual norm 6.559959330184e-03
            2 KSP Residual norm 2.927815370907e-03
            3 KSP Residual norm 4.414378305489e-04
            4 KSP Residual norm 1.261259895261e-04
            5 KSP Residual norm 1.639913030560e-05
            6 KSP Residual norm 2.134357967380e-06
            7 KSP Residual norm 6.760613069100e-17
          1 SNES Function norm 6.867492280866e-04
            0 KSP Residual norm 2.977424249989e-06
            1 KSP Residual norm 3.699416246968e-07
            2 KSP Residual norm 8.447126428579e-08
            3 KSP Residual norm 2.833631246001e-08
            4 KSP Residual norm 5.147530491106e-09
            5 KSP Residual norm 1.094520636202e-09
            6 KSP Residual norm 2.457086187479e-10
            7 KSP Residual norm 7.230574224864e-21
          2 SNES Function norm 4.752621462354e-11
      5 TS dt 0.00847458 time 0.0423729
          0 SNES Function norm 1.045055044742e+01
            0 KSP Residual norm 6.369115842069e-02
            1 KSP Residual norm 6.568022679059e-03
            2 KSP Residual norm 2.908220767658e-03
            3 KSP Residual norm 4.416409177985e-04
            4 KSP Residual norm 1.245501371995e-04
            5 KSP Residual norm 1.638803042750e-05
            6 KSP Residual norm 2.127621812336e-06
            7 KSP Residual norm 7.898425027421e-17
          1 SNES Function norm 6.757059470783e-04
            0 KSP Residual norm 2.940003517446e-06
            1 KSP Residual norm 3.650168880980e-07
            2 KSP Residual norm 8.312328182674e-08
            3 KSP Residual norm 2.790560484320e-08
            4 KSP Residual norm 5.053653328311e-09
            5 KSP Residual norm 1.076108839046e-09
            6 KSP Residual norm 2.402925346880e-10
            7 KSP Residual norm 3.741394720880e-21
          2 SNES Function norm 4.666130700591e-11
      6 TS dt 0.00847458 time 0.0508475
          0 SNES Function norm 1.045184767808e+01
            0 KSP Residual norm 6.378387343082e-02
            1 KSP Residual norm 6.576008998535e-03
            2 KSP Residual norm 2.888789584447e-03
            3 KSP Residual norm 4.418385875472e-04
            4 KSP Residual norm 1.230117980237e-04
            5 KSP Residual norm 1.637733246166e-05
            6 KSP Residual norm 2.121315990116e-06
            7 KSP Residual norm 9.102989092310e-17
          1 SNES Function norm 6.649269539423e-04
            0 KSP Residual norm 2.903304720284e-06
            1 KSP Residual norm 3.601913872254e-07
            2 KSP Residual norm 8.180453791505e-08
            3 KSP Residual norm 2.748442911816e-08
            4 KSP Residual norm 4.961868923662e-09
            5 KSP Residual norm 1.058171714893e-09
            6 KSP Residual norm 2.350195851927e-10
            7 KSP Residual norm 4.964834968715e-21
          2 SNES Function norm 4.471025129852e-11
      7 TS dt 0.00847458 time 0.059322
          0 SNES Function norm 1.045312440769e+01
            0 KSP Residual norm 6.387585481648e-02
            1 KSP Residual norm 6.583919661259e-03
            2 KSP Residual norm 2.869523271170e-03
            3 KSP Residual norm 4.420308468558e-04
            4 KSP Residual norm 1.215101490543e-04
            5 KSP Residual norm 1.636701041839e-05
            6 KSP Residual norm 2.115398352263e-06
            7 KSP Residual norm 6.417124125528e-17
          1 SNES Function norm 6.544038785722e-04
            0 KSP Residual norm 2.867308940893e-06
            1 KSP Residual norm 3.554624318597e-07
            2 KSP Residual norm 8.051422770910e-08
            3 KSP Residual norm 2.707250399844e-08
            4 KSP Residual norm 4.872125443546e-09
            5 KSP Residual norm 1.040692056328e-09
            6 KSP Residual norm 2.298855614716e-10
            7 KSP Residual norm 3.140670460028e-21
          2 SNES Function norm 4.058847439815e-11
      8 TS dt 0.00847458 time 0.0677966
          0 SNES Function norm 1.045438111843e+01
            0 KSP Residual norm 6.396711081541e-02
            1 KSP Residual norm 6.591755999885e-03
            2 KSP Residual norm 2.850423085515e-03
            3 KSP Residual norm 4.422177075225e-04
            4 KSP Residual norm 1.200443665787e-04
            5 KSP Residual norm 1.635703950965e-05
            6 KSP Residual norm 2.109829761879e-06
            7 KSP Residual norm 6.553582414264e-17
          1 SNES Function norm 6.441286836585e-04
            0 KSP Residual norm 2.831997900324e-06
            1 KSP Residual norm 3.508274228910e-07
            2 KSP Residual norm 7.925155398311e-08
            3 KSP Residual norm 2.666955853517e-08
            4 KSP Residual norm 4.784371891515e-09
            5 KSP Residual norm 1.023653459714e-09
            6 KSP Residual norm 2.248862935554e-10
            7 KSP Residual norm 3.952784367220e-21
          2 SNES Function norm 4.092050026541e-11
      9 TS dt 0.00847458 time 0.0762712
          0 SNES Function norm 1.045561827745e+01
            0 KSP Residual norm 6.405764955798e-02
            1 KSP Residual norm 6.599519308575e-03
            2 KSP Residual norm 2.831490102359e-03
            3 KSP Residual norm 4.423991856931e-04
            4 KSP Residual norm 1.186136279928e-04
            5 KSP Residual norm 1.634739610156e-05
            6 KSP Residual norm 2.104573899619e-06
            7 KSP Residual norm 5.913882911408e-17
          1 SNES Function norm 6.340936409061e-04
            0 KSP Residual norm 2.797353906304e-06
            1 KSP Residual norm 3.462838484911e-07
            2 KSP Residual norm 7.801574775723e-08
            3 KSP Residual norm 2.627533147918e-08
            4 KSP Residual norm 4.698557254015e-09
            5 KSP Residual norm 1.007040261358e-09
            6 KSP Residual norm 2.200177803007e-10
            7 KSP Residual norm 3.762004175411e-21
          2 SNES Function norm 4.009143190166e-11
      10 TS dt 0.00847458 time 0.0847458

- Run with the same options on 4 processors with 128 grid points

      $ mpiexec -n 16 ./ex2 -ts_max_steps 10 -ts_monitor -M 128

  Expected output:

      0 TS dt 0.00393701 time 0.
      1 TS dt 0.00393701 time 0.00393701
      2 TS dt 0.00393701 time 0.00787402
      3 TS dt 0.00393701 time 0.011811
      4 TS dt 0.00393701 time 0.015748
      5 TS dt 0.00393701 time 0.019685
      6 TS dt 0.00393701 time 0.023622
      7 TS dt 0.00393701 time 0.0275591
      8 TS dt 0.00393701 time 0.0314961
      9 TS dt 0.00393701 time 0.0354331
      10 TS dt 0.00393701 time 0.0393701

## Nonlinear PDE on a structured grid

WHAT THIS EXAMPLE DEMONSTRATES:

- Handling a 2d structured grid

- Using the nonlinear solvers

- Changing the default linear solver

FURTHER DETAILS:

- Mathematical description of the problem

- main program source code

- physics source code

DO THE FOLLOWING:

- Compile `src/snes/tutorials/ex19.c`

      $ cd petsc/src/snes/tutorials/
      $ make ex19

- Run a 4 processor example with 5 levels of grid refinement, monitor the convergence of the nonlinear and linear solver and examine the exact solver used

      $ mpiexec -n 4 ./ex19 -da_refine 5 -snes_monitor -ksp_monitor -snes_view

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
        0 SNES Function norm 1.036007954337e-02
          0 KSP Residual norm 9.144944502871e-02
          1 KSP Residual norm 2.593759906204e-02
          2 KSP Residual norm 1.669815200495e-02
          3 KSP Residual norm 1.510777951698e-02
          4 KSP Residual norm 1.458401237884e-02
          5 KSP Residual norm 1.418635322926e-02
          6 KSP Residual norm 1.377436725003e-02
          7 KSP Residual norm 1.332236907186e-02
          8 KSP Residual norm 1.288602527920e-02
          9 KSP Residual norm 1.240018288138e-02
         10 KSP Residual norm 1.186798872492e-02
         11 KSP Residual norm 1.126565820237e-02
         12 KSP Residual norm 1.063916289485e-02
         13 KSP Residual norm 9.975377414549e-03
         14 KSP Residual norm 9.354874256053e-03
         15 KSP Residual norm 8.779095086508e-03
         16 KSP Residual norm 8.257220099779e-03
         17 KSP Residual norm 7.721504294489e-03
         18 KSP Residual norm 7.165931362294e-03
         19 KSP Residual norm 6.614579158468e-03
         20 KSP Residual norm 6.069852590203e-03
         21 KSP Residual norm 5.532583715678e-03
         22 KSP Residual norm 5.012542062575e-03
         23 KSP Residual norm 4.469698743551e-03
         24 KSP Residual norm 3.946112491958e-03
         25 KSP Residual norm 3.431223373793e-03
         26 KSP Residual norm 2.968213186086e-03
         27 KSP Residual norm 2.622089571760e-03
         28 KSP Residual norm 2.418543969985e-03
         29 KSP Residual norm 2.310824403854e-03
         30 KSP Residual norm 2.236309852146e-03
         31 KSP Residual norm 2.197075836614e-03
         32 KSP Residual norm 2.184744877441e-03
         33 KSP Residual norm 2.184139801213e-03
         34 KSP Residual norm 2.177357542486e-03
         35 KSP Residual norm 2.165221418018e-03
         36 KSP Residual norm 2.150741735309e-03
         37 KSP Residual norm 2.135576712445e-03
         38 KSP Residual norm 2.119308727352e-03
         39 KSP Residual norm 2.106838711740e-03
         40 KSP Residual norm 2.095968914022e-03
         41 KSP Residual norm 2.087204866778e-03
         42 KSP Residual norm 2.075535849832e-03
         43 KSP Residual norm 2.056901852426e-03
         44 KSP Residual norm 2.026534413197e-03
         45 KSP Residual norm 1.986848633315e-03
         46 KSP Residual norm 1.940721994942e-03
         47 KSP Residual norm 1.883144499412e-03
         48 KSP Residual norm 1.803002891048e-03
         49 KSP Residual norm 1.710431551202e-03
         50 KSP Residual norm 1.590925854814e-03
         51 KSP Residual norm 1.451082323211e-03
         52 KSP Residual norm 1.301330299614e-03
         53 KSP Residual norm 1.182230084682e-03
         54 KSP Residual norm 1.107680018677e-03
         55 KSP Residual norm 1.066507531910e-03
         56 KSP Residual norm 1.042227387049e-03
         57 KSP Residual norm 1.018322560203e-03
         58 KSP Residual norm 9.885570705392e-04
         59 KSP Residual norm 9.548240372245e-04
         60 KSP Residual norm 9.201938192905e-04
         61 KSP Residual norm 9.067345082392e-04
         62 KSP Residual norm 9.048664588409e-04
         63 KSP Residual norm 9.048625048106e-04
         64 KSP Residual norm 9.045786478382e-04
         65 KSP Residual norm 9.025111529622e-04
         66 KSP Residual norm 8.966677177120e-04
         67 KSP Residual norm 8.874037824397e-04
         68 KSP Residual norm 8.789188093430e-04
         69 KSP Residual norm 8.717007141537e-04
         70 KSP Residual norm 8.635394193756e-04
         71 KSP Residual norm 8.549188412543e-04
         72 KSP Residual norm 8.449007643802e-04
         73 KSP Residual norm 8.306833639556e-04
         74 KSP Residual norm 8.180866477839e-04
         75 KSP Residual norm 8.062571853623e-04
         76 KSP Residual norm 7.936883824218e-04
         77 KSP Residual norm 7.780490917915e-04
         78 KSP Residual norm 7.615878046973e-04
         79 KSP Residual norm 7.442269316420e-04
         80 KSP Residual norm 7.232115811673e-04
         81 KSP Residual norm 6.988082322432e-04
         82 KSP Residual norm 6.683230529966e-04
         83 KSP Residual norm 6.362177125994e-04
         84 KSP Residual norm 5.971822394607e-04
         85 KSP Residual norm 5.563666831991e-04
         86 KSP Residual norm 5.122378485297e-04
         87 KSP Residual norm 4.641141819361e-04
         88 KSP Residual norm 4.127674620013e-04
         89 KSP Residual norm 3.767940694930e-04
         90 KSP Residual norm 3.464891353455e-04
         91 KSP Residual norm 3.328326373461e-04
         92 KSP Residual norm 3.152057745476e-04
         93 KSP Residual norm 3.046565182399e-04
         94 KSP Residual norm 3.006541516591e-04
         95 KSP Residual norm 2.987345416223e-04
         96 KSP Residual norm 2.969106149950e-04
         97 KSP Residual norm 2.959679010652e-04
         98 KSP Residual norm 2.951910867025e-04
         99 KSP Residual norm 2.941123180589e-04
        100 KSP Residual norm 2.926765604512e-04
        101 KSP Residual norm 2.900410156058e-04
        102 KSP Residual norm 2.863626926022e-04
        103 KSP Residual norm 2.810952824275e-04
        104 KSP Residual norm 2.755878113867e-04
        105 KSP Residual norm 2.712557488209e-04
        106 KSP Residual norm 2.679296794391e-04
        107 KSP Residual norm 2.642026368303e-04
        108 KSP Residual norm 2.602137324935e-04
        109 KSP Residual norm 2.541382700234e-04
        110 KSP Residual norm 2.454344737954e-04
        111 KSP Residual norm 2.335628910244e-04
        112 KSP Residual norm 2.168004908802e-04
        113 KSP Residual norm 1.993544334742e-04
        114 KSP Residual norm 1.812661079898e-04
        115 KSP Residual norm 1.614155068832e-04
        116 KSP Residual norm 1.450176652465e-04
        117 KSP Residual norm 1.271814682476e-04
        118 KSP Residual norm 1.114893752683e-04
        119 KSP Residual norm 1.016403410116e-04
        120 KSP Residual norm 9.221659791770e-05
        121 KSP Residual norm 8.791115165789e-05
        122 KSP Residual norm 8.529594818471e-05
        123 KSP Residual norm 8.439823024838e-05
        124 KSP Residual norm 8.432330233590e-05
        125 KSP Residual norm 8.432313969703e-05
        126 KSP Residual norm 8.431433381217e-05
        127 KSP Residual norm 8.424348778495e-05
        128 KSP Residual norm 8.382806777182e-05
        129 KSP Residual norm 8.337135217553e-05
        130 KSP Residual norm 8.306671396769e-05
        131 KSP Residual norm 8.299046396158e-05
        132 KSP Residual norm 8.298022748488e-05
        133 KSP Residual norm 8.296556620174e-05
        134 KSP Residual norm 8.293318221137e-05
        135 KSP Residual norm 8.289997195859e-05
        136 KSP Residual norm 8.288650847461e-05
        137 KSP Residual norm 8.287793944348e-05
        138 KSP Residual norm 8.282009713924e-05
        139 KSP Residual norm 8.231936644200e-05
        140 KSP Residual norm 8.092917384457e-05
        141 KSP Residual norm 7.810875275548e-05
        142 KSP Residual norm 7.372335915736e-05
        143 KSP Residual norm 6.920873807564e-05
        144 KSP Residual norm 6.510777151187e-05
        145 KSP Residual norm 6.142132751787e-05
        146 KSP Residual norm 5.816161902635e-05
        147 KSP Residual norm 5.516213050219e-05
        148 KSP Residual norm 5.242284932630e-05
        149 KSP Residual norm 4.986057648037e-05
        150 KSP Residual norm 4.733288055568e-05
        151 KSP Residual norm 4.601506226246e-05
        152 KSP Residual norm 4.523121336508e-05
        153 KSP Residual norm 4.507733287029e-05
        154 KSP Residual norm 4.507097170108e-05
        155 KSP Residual norm 4.506961467378e-05
        156 KSP Residual norm 4.505696152433e-05
        157 KSP Residual norm 4.501836545704e-05
        158 KSP Residual norm 4.495626487199e-05
        159 KSP Residual norm 4.490012973376e-05
        160 KSP Residual norm 4.480136241466e-05
        161 KSP Residual norm 4.458312996682e-05
        162 KSP Residual norm 4.431767440291e-05
        163 KSP Residual norm 4.407964983727e-05
        164 KSP Residual norm 4.383992113924e-05
        165 KSP Residual norm 4.357188317772e-05
        166 KSP Residual norm 4.319522376980e-05
        167 KSP Residual norm 4.261206561683e-05
        168 KSP Residual norm 4.205899613192e-05
        169 KSP Residual norm 4.138430977164e-05
        170 KSP Residual norm 4.047346042359e-05
        171 KSP Residual norm 3.933307539335e-05
        172 KSP Residual norm 3.819767627834e-05
        173 KSP Residual norm 3.702517997681e-05
        174 KSP Residual norm 3.583191921804e-05
        175 KSP Residual norm 3.458798761674e-05
        176 KSP Residual norm 3.316083374306e-05
        177 KSP Residual norm 3.173967896731e-05
        178 KSP Residual norm 3.016354626802e-05
        179 KSP Residual norm 2.866779750173e-05
        180 KSP Residual norm 2.702938196877e-05
        181 KSP Residual norm 2.618361138750e-05
        182 KSP Residual norm 2.522495437254e-05
        183 KSP Residual norm 2.426023897276e-05
        184 KSP Residual norm 2.355948721907e-05
        185 KSP Residual norm 2.319684487218e-05
        186 KSP Residual norm 2.289784420766e-05
        187 KSP Residual norm 2.267598687625e-05
        188 KSP Residual norm 2.240641749204e-05
        189 KSP Residual norm 2.212551730336e-05
        190 KSP Residual norm 2.170264854588e-05
        191 KSP Residual norm 2.112756030054e-05
        192 KSP Residual norm 2.038822399814e-05
        193 KSP Residual norm 1.962951220216e-05
        194 KSP Residual norm 1.884493949304e-05
        195 KSP Residual norm 1.799734026963e-05
        196 KSP Residual norm 1.722254569823e-05
        197 KSP Residual norm 1.660423842819e-05
        198 KSP Residual norm 1.621056066730e-05
        199 KSP Residual norm 1.591018158958e-05
        200 KSP Residual norm 1.557926981647e-05
        201 KSP Residual norm 1.510189268164e-05
        202 KSP Residual norm 1.440759642876e-05
        203 KSP Residual norm 1.349458967348e-05
        204 KSP Residual norm 1.240308276374e-05
        205 KSP Residual norm 1.118091740362e-05
        206 KSP Residual norm 9.955874799398e-06
        207 KSP Residual norm 8.667314210234e-06
        208 KSP Residual norm 7.389939064823e-06
        209 KSP Residual norm 6.261620050378e-06
        210 KSP Residual norm 5.246555512523e-06
        211 KSP Residual norm 4.721004890241e-06
        212 KSP Residual norm 4.239837116741e-06
        213 KSP Residual norm 3.816477467422e-06
        214 KSP Residual norm 3.501683693279e-06
        215 KSP Residual norm 3.305190215185e-06
        216 KSP Residual norm 3.206138813817e-06
        217 KSP Residual norm 3.174323738414e-06
        218 KSP Residual norm 3.169528835126e-06
        219 KSP Residual norm 3.169521851846e-06
        220 KSP Residual norm 3.165241221321e-06
        221 KSP Residual norm 3.145015122355e-06
        222 KSP Residual norm 3.096044377523e-06
        223 KSP Residual norm 3.018842023098e-06
        224 KSP Residual norm 2.964634266861e-06
        225 KSP Residual norm 2.957729966340e-06
        226 KSP Residual norm 2.953877433705e-06
        227 KSP Residual norm 2.925464755647e-06
        228 KSP Residual norm 2.868821700731e-06
        229 KSP Residual norm 2.782027517577e-06
        230 KSP Residual norm 2.646127535134e-06
        231 KSP Residual norm 2.482650898676e-06
        232 KSP Residual norm 2.309998463210e-06
        233 KSP Residual norm 2.154086486854e-06
        234 KSP Residual norm 2.002548899717e-06
        235 KSP Residual norm 1.885163787351e-06
        236 KSP Residual norm 1.820671950047e-06
        237 KSP Residual norm 1.781332450628e-06
        238 KSP Residual norm 1.751510777513e-06
        239 KSP Residual norm 1.723392579686e-06
        240 KSP Residual norm 1.694083934428e-06
        241 KSP Residual norm 1.677189950467e-06
        242 KSP Residual norm 1.673111374168e-06
        243 KSP Residual norm 1.671932556435e-06
        244 KSP Residual norm 1.670372344826e-06
        245 KSP Residual norm 1.668281814293e-06
        246 KSP Residual norm 1.664401756910e-06
        247 KSP Residual norm 1.655699903087e-06
        248 KSP Residual norm 1.644879786465e-06
        249 KSP Residual norm 1.638313410510e-06
        250 KSP Residual norm 1.634433920669e-06
        251 KSP Residual norm 1.632288417000e-06
        252 KSP Residual norm 1.630803349524e-06
        253 KSP Residual norm 1.629108012046e-06
        254 KSP Residual norm 1.625738527055e-06
        255 KSP Residual norm 1.620864125655e-06
        256 KSP Residual norm 1.616268872661e-06
        257 KSP Residual norm 1.611801733029e-06
        258 KSP Residual norm 1.602497312803e-06
        259 KSP Residual norm 1.575738098262e-06
        260 KSP Residual norm 1.519104509227e-06
        261 KSP Residual norm 1.431857168315e-06
        262 KSP Residual norm 1.338744222187e-06
        263 KSP Residual norm 1.262050256308e-06
        264 KSP Residual norm 1.193091353420e-06
        265 KSP Residual norm 1.139031579299e-06
        266 KSP Residual norm 1.088471447383e-06
        267 KSP Residual norm 1.035130328689e-06
        268 KSP Residual norm 9.777098693277e-07
        269 KSP Residual norm 9.155430511512e-07
        270 KSP Residual norm 8.568922899017e-07
        1 SNES Function norm 1.178751989614e-06
          0 KSP Residual norm 1.748815715091e-06
          1 KSP Residual norm 1.624620439395e-06
          2 KSP Residual norm 1.456422465392e-06
          3 KSP Residual norm 1.326030517472e-06
          4 KSP Residual norm 1.134584001300e-06
          5 KSP Residual norm 9.824370585989e-07
          6 KSP Residual norm 8.882499873515e-07
          7 KSP Residual norm 8.249609129314e-07
          8 KSP Residual norm 7.838632525267e-07
          9 KSP Residual norm 7.558367537184e-07
         10 KSP Residual norm 7.352641428514e-07
         11 KSP Residual norm 7.177890029352e-07
         12 KSP Residual norm 7.027407189224e-07
         13 KSP Residual norm 6.870092629142e-07
         14 KSP Residual norm 6.712782681272e-07
         15 KSP Residual norm 6.556064942447e-07
         16 KSP Residual norm 6.413026840450e-07
         17 KSP Residual norm 6.250492096162e-07
         18 KSP Residual norm 6.087611627271e-07
         19 KSP Residual norm 5.930996641661e-07
         20 KSP Residual norm 5.781788025672e-07
         21 KSP Residual norm 5.610549351106e-07
         22 KSP Residual norm 5.401956055125e-07
         23 KSP Residual norm 5.168528059550e-07
         24 KSP Residual norm 4.913547400553e-07
         25 KSP Residual norm 4.653072018102e-07
         26 KSP Residual norm 4.457206633372e-07
         27 KSP Residual norm 4.263897743643e-07
         28 KSP Residual norm 4.072207343179e-07
         29 KSP Residual norm 3.820129426326e-07
         30 KSP Residual norm 3.524926057079e-07
         31 KSP Residual norm 3.348441637200e-07
         32 KSP Residual norm 3.208234358783e-07
         33 KSP Residual norm 3.080653563509e-07
         34 KSP Residual norm 2.969658623379e-07
         35 KSP Residual norm 2.873838557550e-07
         36 KSP Residual norm 2.784716738215e-07
         37 KSP Residual norm 2.682605079585e-07
         38 KSP Residual norm 2.582627271510e-07
         39 KSP Residual norm 2.471851529652e-07
         40 KSP Residual norm 2.364422933814e-07
         41 KSP Residual norm 2.262205960981e-07
         42 KSP Residual norm 2.177174676790e-07
         43 KSP Residual norm 2.119314681575e-07
         44 KSP Residual norm 2.086939643300e-07
         45 KSP Residual norm 2.069589867774e-07
         46 KSP Residual norm 2.051651483193e-07
         47 KSP Residual norm 2.028925351652e-07
         48 KSP Residual norm 1.994371389840e-07
         49 KSP Residual norm 1.960711107273e-07
         50 KSP Residual norm 1.934148114236e-07
         51 KSP Residual norm 1.905853168179e-07
         52 KSP Residual norm 1.867410083506e-07
         53 KSP Residual norm 1.820712301605e-07
         54 KSP Residual norm 1.769911631032e-07
         55 KSP Residual norm 1.726229506451e-07
         56 KSP Residual norm 1.684625473534e-07
         57 KSP Residual norm 1.621405677874e-07
         58 KSP Residual norm 1.531432035392e-07
         59 KSP Residual norm 1.424389275969e-07
         60 KSP Residual norm 1.310183104254e-07
         61 KSP Residual norm 1.242656204932e-07
         62 KSP Residual norm 1.179240873971e-07
         63 KSP Residual norm 1.123101522446e-07
         64 KSP Residual norm 1.054132720446e-07
         65 KSP Residual norm 9.849626843795e-08
         66 KSP Residual norm 9.197247865719e-08
         67 KSP Residual norm 8.638142369566e-08
         68 KSP Residual norm 8.305351990631e-08
         69 KSP Residual norm 8.136422492034e-08
         70 KSP Residual norm 8.002361575492e-08
         71 KSP Residual norm 7.866116430480e-08
         72 KSP Residual norm 7.722692892725e-08
         73 KSP Residual norm 7.544381224969e-08
         74 KSP Residual norm 7.354404089853e-08
         75 KSP Residual norm 7.119435912763e-08
         76 KSP Residual norm 6.867518671189e-08
         77 KSP Residual norm 6.602411006203e-08
         78 KSP Residual norm 6.337351313821e-08
         79 KSP Residual norm 6.080224058330e-08
         80 KSP Residual norm 5.890231340688e-08
         81 KSP Residual norm 5.776735028390e-08
         82 KSP Residual norm 5.681661116681e-08
         83 KSP Residual norm 5.569418042042e-08
         84 KSP Residual norm 5.415755708381e-08
         85 KSP Residual norm 5.233260646242e-08
         86 KSP Residual norm 5.021186255249e-08
         87 KSP Residual norm 4.793874560013e-08
         88 KSP Residual norm 4.567033260035e-08
         89 KSP Residual norm 4.349813734988e-08
         90 KSP Residual norm 4.106532397968e-08
         91 KSP Residual norm 3.953311916974e-08
         92 KSP Residual norm 3.782549069823e-08
         93 KSP Residual norm 3.618014409136e-08
         94 KSP Residual norm 3.473408466117e-08
         95 KSP Residual norm 3.295298494632e-08
         96 KSP Residual norm 3.116362368283e-08
         97 KSP Residual norm 2.964729931235e-08
         98 KSP Residual norm 2.843092504497e-08
         99 KSP Residual norm 2.752698726769e-08
        100 KSP Residual norm 2.677449961663e-08
        101 KSP Residual norm 2.623140990700e-08
        102 KSP Residual norm 2.574444690034e-08
        103 KSP Residual norm 2.522405798326e-08
        104 KSP Residual norm 2.457253380175e-08
        105 KSP Residual norm 2.370264120442e-08
        106 KSP Residual norm 2.262668411404e-08
        107 KSP Residual norm 2.152259489644e-08
        108 KSP Residual norm 2.036137945628e-08
        109 KSP Residual norm 1.909454196392e-08
        110 KSP Residual norm 1.787445336551e-08
        111 KSP Residual norm 1.667394027353e-08
        112 KSP Residual norm 1.559924659359e-08
        113 KSP Residual norm 1.481283115471e-08
        114 KSP Residual norm 1.435158522296e-08
        115 KSP Residual norm 1.414878343957e-08
        116 KSP Residual norm 1.407393287977e-08
        117 KSP Residual norm 1.404447798640e-08
        118 KSP Residual norm 1.403314447128e-08
        119 KSP Residual norm 1.402765026958e-08
        120 KSP Residual norm 1.402395814514e-08
        121 KSP Residual norm 1.402351651527e-08
        122 KSP Residual norm 1.402351613398e-08
        123 KSP Residual norm 1.402327634838e-08
        124 KSP Residual norm 1.402226170240e-08
        125 KSP Residual norm 1.402020560080e-08
        126 KSP Residual norm 1.401585203411e-08
        127 KSP Residual norm 1.400213674922e-08
        128 KSP Residual norm 1.396738912660e-08
        129 KSP Residual norm 1.392184633617e-08
        130 KSP Residual norm 1.386006552773e-08
        131 KSP Residual norm 1.378213258746e-08
        132 KSP Residual norm 1.368010395178e-08
        133 KSP Residual norm 1.353666156547e-08
        134 KSP Residual norm 1.329698812650e-08
        135 KSP Residual norm 1.293880270293e-08
        136 KSP Residual norm 1.237318583853e-08
        137 KSP Residual norm 1.179161581684e-08
        138 KSP Residual norm 1.134206743886e-08
        139 KSP Residual norm 1.099888749064e-08
        140 KSP Residual norm 1.066259659918e-08
        141 KSP Residual norm 1.032068623282e-08
        142 KSP Residual norm 9.954236804205e-09
        143 KSP Residual norm 9.560912215756e-09
        144 KSP Residual norm 9.145545126143e-09
        145 KSP Residual norm 8.738420927086e-09
        146 KSP Residual norm 8.409260212909e-09
        147 KSP Residual norm 8.127321870356e-09
        148 KSP Residual norm 7.898748854589e-09
        149 KSP Residual norm 7.703865149553e-09
        150 KSP Residual norm 7.486870703879e-09
        151 KSP Residual norm 7.330560146692e-09
        152 KSP Residual norm 7.128689079801e-09
        153 KSP Residual norm 6.964286032888e-09
        154 KSP Residual norm 6.785137395884e-09
        155 KSP Residual norm 6.640351575110e-09
        156 KSP Residual norm 6.511156528025e-09
        157 KSP Residual norm 6.423541298314e-09
        158 KSP Residual norm 6.345751133879e-09
        159 KSP Residual norm 6.280901533539e-09
        160 KSP Residual norm 6.207943324992e-09
        161 KSP Residual norm 6.142867578714e-09
        162 KSP Residual norm 6.087418250155e-09
        163 KSP Residual norm 6.034648495317e-09
        164 KSP Residual norm 5.977088198390e-09
        165 KSP Residual norm 5.912054132468e-09
        166 KSP Residual norm 5.836856906294e-09
        167 KSP Residual norm 5.765552516298e-09
        168 KSP Residual norm 5.679756378903e-09
        169 KSP Residual norm 5.563707770725e-09
        170 KSP Residual norm 5.405011361483e-09
        171 KSP Residual norm 5.189986018558e-09
        172 KSP Residual norm 4.908102634804e-09
        173 KSP Residual norm 4.706174551357e-09
        174 KSP Residual norm 4.546685413987e-09
        175 KSP Residual norm 4.435455313583e-09
        176 KSP Residual norm 4.332523827655e-09
        177 KSP Residual norm 4.203498282754e-09
        178 KSP Residual norm 4.058340805179e-09
        179 KSP Residual norm 3.912110698046e-09
        180 KSP Residual norm 3.754156543395e-09
        181 KSP Residual norm 3.673186018443e-09
        182 KSP Residual norm 3.613029839620e-09
        183 KSP Residual norm 3.561927158378e-09
        184 KSP Residual norm 3.500712553539e-09
        185 KSP Residual norm 3.426672494749e-09
        186 KSP Residual norm 3.351413827965e-09
        187 KSP Residual norm 3.271649485418e-09
        188 KSP Residual norm 3.188154724333e-09
        189 KSP Residual norm 3.100224644403e-09
        190 KSP Residual norm 3.010790734288e-09
        191 KSP Residual norm 2.899860376120e-09
        192 KSP Residual norm 2.795307867707e-09
        193 KSP Residual norm 2.656859285349e-09
        194 KSP Residual norm 2.478450141125e-09
        195 KSP Residual norm 2.312417968425e-09
        196 KSP Residual norm 2.135819348150e-09
        197 KSP Residual norm 1.987345509540e-09
        198 KSP Residual norm 1.882308219422e-09
        199 KSP Residual norm 1.824720724119e-09
        200 KSP Residual norm 1.793394737731e-09
        201 KSP Residual norm 1.777603206282e-09
        202 KSP Residual norm 1.767124708984e-09
        203 KSP Residual norm 1.759460100944e-09
        204 KSP Residual norm 1.754016726123e-09
        205 KSP Residual norm 1.750274824009e-09
        206 KSP Residual norm 1.747607840035e-09
        207 KSP Residual norm 1.744480855293e-09
        208 KSP Residual norm 1.741655356119e-09
        209 KSP Residual norm 1.739290578624e-09
        210 KSP Residual norm 1.734739843580e-09
        211 KSP Residual norm 1.731591096986e-09
        212 KSP Residual norm 1.728201695680e-09
        213 KSP Residual norm 1.723790864899e-09
        214 KSP Residual norm 1.713216135283e-09
        215 KSP Residual norm 1.699328282646e-09
        216 KSP Residual norm 1.678451518644e-09
        217 KSP Residual norm 1.657901783704e-09
        218 KSP Residual norm 1.635934083952e-09
        219 KSP Residual norm 1.612202184752e-09
        220 KSP Residual norm 1.567594145713e-09
        221 KSP Residual norm 1.512642314015e-09
        222 KSP Residual norm 1.457761603944e-09
        223 KSP Residual norm 1.408111435897e-09
        224 KSP Residual norm 1.356076746727e-09
        225 KSP Residual norm 1.307962107684e-09
        226 KSP Residual norm 1.266646107226e-09
        227 KSP Residual norm 1.231345481628e-09
        228 KSP Residual norm 1.187725733507e-09
        229 KSP Residual norm 1.141044747451e-09
        230 KSP Residual norm 1.099737269841e-09
        231 KSP Residual norm 1.066786397851e-09
        232 KSP Residual norm 1.030129878172e-09
        233 KSP Residual norm 9.927434935483e-10
        234 KSP Residual norm 9.473773325131e-10
        235 KSP Residual norm 9.089690854957e-10
        236 KSP Residual norm 8.759516453077e-10
        237 KSP Residual norm 8.535012664712e-10
        238 KSP Residual norm 8.308754136837e-10
        239 KSP Residual norm 8.082501666452e-10
        240 KSP Residual norm 7.754022320857e-10
        241 KSP Residual norm 7.572112123056e-10
        242 KSP Residual norm 7.442389885537e-10
        243 KSP Residual norm 7.283799305392e-10
        244 KSP Residual norm 7.073231969200e-10
        245 KSP Residual norm 6.852558048466e-10
        246 KSP Residual norm 6.637193841945e-10
        247 KSP Residual norm 6.457535438239e-10
        248 KSP Residual norm 6.348852218182e-10
        249 KSP Residual norm 6.257254477629e-10
        250 KSP Residual norm 6.158564534747e-10
        251 KSP Residual norm 6.053446723415e-10
        252 KSP Residual norm 5.965943232727e-10
        253 KSP Residual norm 5.889879966631e-10
        254 KSP Residual norm 5.798723091489e-10
        255 KSP Residual norm 5.693837266918e-10
        256 KSP Residual norm 5.572444317590e-10
        257 KSP Residual norm 5.444694016783e-10
        258 KSP Residual norm 5.294254215977e-10
        259 KSP Residual norm 5.111579131674e-10
        260 KSP Residual norm 4.907812169770e-10
        261 KSP Residual norm 4.698943704786e-10
        262 KSP Residual norm 4.516736830848e-10
        263 KSP Residual norm 4.344099614072e-10
        264 KSP Residual norm 4.205651998682e-10
        265 KSP Residual norm 4.094400044417e-10
        266 KSP Residual norm 4.007111452005e-10
        267 KSP Residual norm 3.933386697206e-10
        268 KSP Residual norm 3.859019001818e-10
        269 KSP Residual norm 3.778293647154e-10
        270 KSP Residual norm 3.685811905503e-10
        271 KSP Residual norm 3.629152212256e-10
        272 KSP Residual norm 3.564840338194e-10
        273 KSP Residual norm 3.500495329589e-10
        274 KSP Residual norm 3.431898068167e-10
        275 KSP Residual norm 3.353565364446e-10
        276 KSP Residual norm 3.263783927727e-10
        277 KSP Residual norm 3.192763021450e-10
        278 KSP Residual norm 3.122085426690e-10
        279 KSP Residual norm 3.052760168605e-10
        280 KSP Residual norm 2.980518819939e-10
        281 KSP Residual norm 2.912439358242e-10
        282 KSP Residual norm 2.849441584275e-10
        283 KSP Residual norm 2.782808821251e-10
        284 KSP Residual norm 2.710684441319e-10
        285 KSP Residual norm 2.633874526828e-10
        286 KSP Residual norm 2.553129453009e-10
        287 KSP Residual norm 2.463101545110e-10
        288 KSP Residual norm 2.377718932698e-10
        289 KSP Residual norm 2.297278980842e-10
        290 KSP Residual norm 2.216396694908e-10
        291 KSP Residual norm 2.140852462454e-10
        292 KSP Residual norm 2.058870942400e-10
        293 KSP Residual norm 1.971167887405e-10
        294 KSP Residual norm 1.884340894074e-10
        295 KSP Residual norm 1.791613990270e-10
        296 KSP Residual norm 1.688633513441e-10
        297 KSP Residual norm 1.602704727320e-10
        298 KSP Residual norm 1.541392323809e-10
        299 KSP Residual norm 1.504896772477e-10
        300 KSP Residual norm 1.483487862182e-10
        301 KSP Residual norm 1.471763898232e-10
        302 KSP Residual norm 1.458511285335e-10
        303 KSP Residual norm 1.446834701991e-10
        304 KSP Residual norm 1.422626457779e-10
        305 KSP Residual norm 1.383238030475e-10
        306 KSP Residual norm 1.340609360645e-10
        307 KSP Residual norm 1.295566320698e-10
        308 KSP Residual norm 1.254740222524e-10
        309 KSP Residual norm 1.211504994547e-10
        310 KSP Residual norm 1.174647456465e-10
        311 KSP Residual norm 1.136881957093e-10
        312 KSP Residual norm 1.097588329841e-10
        313 KSP Residual norm 1.041565322506e-10
        314 KSP Residual norm 9.702228215912e-11
        315 KSP Residual norm 8.989484119100e-11
        316 KSP Residual norm 8.190664018704e-11
        317 KSP Residual norm 7.558818834442e-11
        318 KSP Residual norm 6.957179676675e-11
        319 KSP Residual norm 6.447734334845e-11
        320 KSP Residual norm 6.116288878725e-11
        321 KSP Residual norm 5.888979447543e-11
        322 KSP Residual norm 5.764019417950e-11
        323 KSP Residual norm 5.691872542062e-11
        324 KSP Residual norm 5.660800677735e-11
        325 KSP Residual norm 5.653667085624e-11
        326 KSP Residual norm 5.653626356788e-11
        327 KSP Residual norm 5.653235239410e-11
        328 KSP Residual norm 5.653229638983e-11
        329 KSP Residual norm 5.651625676369e-11
        330 KSP Residual norm 5.646144096698e-11
        331 KSP Residual norm 5.644627590998e-11
        332 KSP Residual norm 5.644531324883e-11
        333 KSP Residual norm 5.644497420702e-11
        334 KSP Residual norm 5.644493580837e-11
        335 KSP Residual norm 5.644470457969e-11
        336 KSP Residual norm 5.644168745929e-11
        337 KSP Residual norm 5.640053714959e-11
        338 KSP Residual norm 5.626705339217e-11
        339 KSP Residual norm 5.617632737039e-11
        340 KSP Residual norm 5.610418525034e-11
        341 KSP Residual norm 5.596338060631e-11
        342 KSP Residual norm 5.565290829848e-11
        343 KSP Residual norm 5.515768341466e-11
        344 KSP Residual norm 5.442406254737e-11
        345 KSP Residual norm 5.353934703450e-11
        346 KSP Residual norm 5.271979463229e-11
        347 KSP Residual norm 5.159719023218e-11
        348 KSP Residual norm 5.050230595494e-11
        349 KSP Residual norm 4.886604439750e-11
        350 KSP Residual norm 4.701727019984e-11
        351 KSP Residual norm 4.529669235590e-11
        352 KSP Residual norm 4.321338221199e-11
        353 KSP Residual norm 4.146974082227e-11
        354 KSP Residual norm 3.989258303406e-11
        355 KSP Residual norm 3.833287369436e-11
        356 KSP Residual norm 3.696924118282e-11
        357 KSP Residual norm 3.572876582097e-11
        358 KSP Residual norm 3.466908142629e-11
        359 KSP Residual norm 3.362405494859e-11
        360 KSP Residual norm 3.248076664724e-11
        361 KSP Residual norm 3.183961506680e-11
        362 KSP Residual norm 3.105405824388e-11
        363 KSP Residual norm 3.019347608164e-11
        364 KSP Residual norm 2.915901620646e-11
        365 KSP Residual norm 2.819671262899e-11
        366 KSP Residual norm 2.737270822959e-11
        367 KSP Residual norm 2.674106883731e-11
        368 KSP Residual norm 2.627612843956e-11
        369 KSP Residual norm 2.574335369953e-11
        370 KSP Residual norm 2.523339840051e-11
        371 KSP Residual norm 2.465910296336e-11
        372 KSP Residual norm 2.407616725222e-11
        373 KSP Residual norm 2.348589790024e-11
        374 KSP Residual norm 2.305940230387e-11
        375 KSP Residual norm 2.268699192687e-11
        376 KSP Residual norm 2.227499543133e-11
        377 KSP Residual norm 2.188320344640e-11
        378 KSP Residual norm 2.152291481840e-11
        379 KSP Residual norm 2.108875033860e-11
        380 KSP Residual norm 2.062954609947e-11
        381 KSP Residual norm 2.007204894177e-11
        382 KSP Residual norm 1.941423359969e-11
        383 KSP Residual norm 1.872835639533e-11
        384 KSP Residual norm 1.794965148165e-11
        385 KSP Residual norm 1.714082018695e-11
        2 SNES Function norm 1.215516598864e-11
      SNES Object: 4 MPI processes
        type: newtonls
        maximum iterations=50, maximum function evaluations=10000
        tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
        total number of linear solver iterations=655
        total number of function evaluations=3
        norm schedule ALWAYS
        Jacobian is built using colored finite differences on a DMDA
        SNESLineSearch Object: 4 MPI processes
          type: bt
            interpolation: cubic
            alpha=1.000000e-04
          maxlambda=1.000000e+00, minlambda=1.000000e-12
          tolerances: relative=1.000000e-08, absolute=1.000000e-15, lambda=1.000000e-08
          maximum iterations=40
        KSP Object: 4 MPI processes
          type: gmres
            restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
            happy breakdown tolerance 1e-30
          maximum iterations=10000, initial guess is zero
          tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
          left preconditioning
          using PRECONDITIONED norm type for convergence test
        PC Object: 4 MPI processes
          type: bjacobi
            number of blocks = 4
            Local solver information for first block is in the following KSP and PC objects on rank 0:
            Use -ksp_view ::ascii_info_detail to display information for all blocks
          KSP Object: (sub_) 1 MPI process
            type: preonly
            maximum iterations=10000, initial guess is zero
            tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
            left preconditioning
            using NONE norm type for convergence test
          PC Object: (sub_) 1 MPI process
            type: ilu
              out-of-place factorization
              0 levels of fill
              tolerance for zero pivot 2.22045e-14
              matrix ordering: natural
              factor fill ratio given 1., needed 1.
                Factored matrix follows:
                  Mat Object: (sub_) 1 MPI process
                    type: seqaij
                    rows=9604, cols=9604, bs=4
                    package used to perform factorization: petsc
                    total: nonzeros=188944, allocated nonzeros=188944
                      using I-node routines: found 2401 nodes, limit used is 5
            linear system matrix = precond matrix:
            Mat Object: (sub_) 1 MPI process
              type: seqaij
              rows=9604, cols=9604, bs=4
              total: nonzeros=188944, allocated nonzeros=188944
              total number of mallocs used during MatSetValues calls=0
                using I-node routines: found 2401 nodes, limit used is 5
          linear system matrix = precond matrix:
          Mat Object: 4 MPI processes
            type: mpiaij
            rows=37636, cols=37636, bs=4
            total: nonzeros=746512, allocated nonzeros=746512
            total number of mallocs used during MatSetValues calls=0
      Number of SNES iterations = 2

- Run with the same options but use geometric multigrid as the linear solver

      $ mpiexec -n 4 ./ex19 -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type mg

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
        0 SNES Function norm 1.036007954337e-02
          0 KSP Residual norm 2.388589583549e+00
          1 KSP Residual norm 5.715829806981e-01
          2 KSP Residual norm 4.623679005936e-02
          3 KSP Residual norm 1.143381177646e-02
          4 KSP Residual norm 2.015139840224e-03
          5 KSP Residual norm 4.356196119798e-04
          6 KSP Residual norm 4.240953066710e-05
          7 KSP Residual norm 8.848315297175e-06
        1 SNES Function norm 9.854304971115e-06
          0 KSP Residual norm 3.868049496775e-05
          1 KSP Residual norm 7.693574326868e-06
          2 KSP Residual norm 1.059429116239e-06
          3 KSP Residual norm 4.004524784804e-07
          4 KSP Residual norm 1.050186948327e-07
          5 KSP Residual norm 5.073180513583e-08
          6 KSP Residual norm 2.510513776297e-08
          7 KSP Residual norm 1.211886495400e-08
          8 KSP Residual norm 1.911963112131e-09
          9 KSP Residual norm 3.005260864225e-10
        2 SNES Function norm 3.117674497824e-10
          0 KSP Residual norm 3.005042584730e-10
          1 KSP Residual norm 1.120673922713e-10
          2 KSP Residual norm 3.288439453292e-11
          3 KSP Residual norm 5.822504321413e-12
          4 KSP Residual norm 2.486684466178e-12
          5 KSP Residual norm 1.198858055503e-12
          6 KSP Residual norm 6.255669709502e-13
          7 KSP Residual norm 1.544647758005e-13
          8 KSP Residual norm 4.592122224907e-14
          9 KSP Residual norm 4.984149547392e-15
         10 KSP Residual norm 8.905129652955e-16
        3 SNES Function norm 1.045594761851e-14
      SNES Object: 4 MPI processes
        type: newtonls
        maximum iterations=50, maximum function evaluations=10000
        tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
        total number of linear solver iterations=26
        total number of function evaluations=4
        norm schedule ALWAYS
        Jacobian is built using colored finite differences on a DMDA
        SNESLineSearch Object: 4 MPI processes
          type: bt
            interpolation: cubic
            alpha=1.000000e-04
          maxlambda=1.000000e+00, minlambda=1.000000e-12
          tolerances: relative=1.000000e-08, absolute=1.000000e-15, lambda=1.000000e-08
          maximum iterations=40
        KSP Object: 4 MPI processes
          type: gmres
            restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
            happy breakdown tolerance 1e-30
          maximum iterations=10000, initial guess is zero
          tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
          left preconditioning
          using PRECONDITIONED norm type for convergence test
        PC Object: 4 MPI processes
          type: mg
            type is MULTIPLICATIVE, levels=6 cycles=v
              Cycles per PCApply=1
              Not using Galerkin computed coarse grid matrices
          Coarse grid solver -- level 0 -------------------------------
            KSP Object: (mg_coarse_) 4 MPI processes
              type: preonly
              maximum iterations=10000, initial guess is zero
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_coarse_) 4 MPI processes
              type: redundant
                First (color=0) of 4 PCs follows
                KSP Object: (mg_coarse_redundant_) 1 MPI process
                  type: preonly
                  maximum iterations=10000, initial guess is zero
                  tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using NONE norm type for convergence test
                PC Object: (mg_coarse_redundant_) 1 MPI process
                  type: lu
                    out-of-place factorization
                    tolerance for zero pivot 2.22045e-14
                    using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
                    matrix ordering: nd
                    factor fill ratio given 5., needed 1.875
                      Factored matrix follows:
                        Mat Object: (mg_coarse_redundant_) 1 MPI process
                          type: seqaij
                          rows=64, cols=64, bs=4
                          package used to perform factorization: petsc
                          total: nonzeros=1920, allocated nonzeros=1920
                            using I-node routines: found 16 nodes, limit used is 5
                  linear system matrix = precond matrix:
                  Mat Object: 1 MPI process
                    type: seqaij
                    rows=64, cols=64, bs=4
                    total: nonzeros=1024, allocated nonzeros=1024
                    total number of mallocs used during MatSetValues calls=0
                      using I-node routines: found 16 nodes, limit used is 5
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=64, cols=64, bs=4
                total: nonzeros=1024, allocated nonzeros=1024
                total number of mallocs used during MatSetValues calls=0
          Down solver (pre-smoother) on level 1 -------------------------------
            KSP Object: (mg_levels_1_) 4 MPI processes
              type: chebyshev
                Chebyshev polynomial of first kind
                eigenvalue targets used: min 0.148269, max 1.63095
                eigenvalues estimated via gmres: min 0.144902, max 1.48269
                eigenvalues estimated using gmres with transform: [0. 0.1; 0. 1.1]
                KSP Object: (mg_levels_1_esteig_) 4 MPI processes
                  type: gmres
                    restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
                    happy breakdown tolerance 1e-30
                  maximum iterations=10, initial guess is zero
                  tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using PRECONDITIONED norm type for convergence test
                estimating eigenvalues using a noisy random number generated right-hand side
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_1_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=196, cols=196, bs=4
                total: nonzeros=3472, allocated nonzeros=3472
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 2 -------------------------------
            KSP Object: (mg_levels_2_) 4 MPI processes
              type: chebyshev
                Chebyshev polynomial of first kind
                eigenvalue targets used: min 0.149178, max 1.64096
                eigenvalues estimated via gmres: min 0.0843938, max 1.49178
                eigenvalues estimated using gmres with transform: [0. 0.1; 0. 1.1]
                KSP Object: (mg_levels_2_esteig_) 4 MPI processes
                  type: gmres
                    restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
                    happy breakdown tolerance 1e-30
                  maximum iterations=10, initial guess is zero
                  tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using PRECONDITIONED norm type for convergence test
                estimating eigenvalues using a noisy random number generated right-hand side
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_2_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=676, cols=676, bs=4
                total: nonzeros=12688, allocated nonzeros=12688
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 3 -------------------------------
            KSP Object: (mg_levels_3_) 4 MPI processes
              type: chebyshev
                Chebyshev polynomial of first kind
                eigenvalue targets used: min 0.146454, max 1.61099
                eigenvalues estimated via gmres: min 0.0659153, max 1.46454
                eigenvalues estimated using gmres with transform: [0. 0.1; 0. 1.1]
                KSP Object: (mg_levels_3_esteig_) 4 MPI processes
                  type: gmres
                    restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
                    happy breakdown tolerance 1e-30
                  maximum iterations=10, initial guess is zero
                  tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using PRECONDITIONED norm type for convergence test
                estimating eigenvalues using a noisy random number generated right-hand side
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_3_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=2500, cols=2500, bs=4
                total: nonzeros=48400, allocated nonzeros=48400
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 4 -------------------------------
            KSP Object: (mg_levels_4_) 4 MPI processes
              type: chebyshev
                Chebyshev polynomial of first kind
                eigenvalue targets used: min 0.141089, max 1.55197
                eigenvalues estimated via gmres: min 0.044097, max 1.41089
                eigenvalues estimated using gmres with transform: [0. 0.1; 0. 1.1]
                KSP Object: (mg_levels_4_esteig_) 4 MPI processes
                  type: gmres
                    restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
                    happy breakdown tolerance 1e-30
                  maximum iterations=10, initial guess is zero
                  tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using PRECONDITIONED norm type for convergence test
                estimating eigenvalues using a noisy random number generated right-hand side
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_4_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=9604, cols=9604, bs=4
                total: nonzeros=188944, allocated nonzeros=188944
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 5 -------------------------------
            KSP Object: (mg_levels_5_) 4 MPI processes
              type: chebyshev
                Chebyshev polynomial of first kind
                eigenvalue targets used: min 0.127956, max 1.40751
                eigenvalues estimated via gmres: min 0.0380398, max 1.27956
                eigenvalues estimated using gmres with transform: [0. 0.1; 0. 1.1]
                KSP Object: (mg_levels_5_esteig_) 4 MPI processes
                  type: gmres
                    restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
                    happy breakdown tolerance 1e-30
                  maximum iterations=10, initial guess is zero
                  tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using PRECONDITIONED norm type for convergence test
                estimating eigenvalues using a noisy random number generated right-hand side
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_5_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=37636, cols=37636, bs=4
                total: nonzeros=746512, allocated nonzeros=746512
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          linear system matrix = precond matrix:
          Mat Object: 4 MPI processes
            type: mpiaij
            rows=37636, cols=37636, bs=4
            total: nonzeros=746512, allocated nonzeros=746512
            total number of mallocs used during MatSetValues calls=0
      Number of SNES iterations = 3

  Note this requires many fewer iterations than the default solver

- Run with the same options but use algebraic multigrid (hypre’s BoomerAMG) as the linear solver

      $ mpiexec -n 4 ./ex19 -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type hypre

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
        0 SNES Function norm 1.036007954337e-02
        1 SNES Function norm 3.201130549705e-06
        2 SNES Function norm 2.738200342766e-10
        3 SNES Function norm 1.086838199638e-14
      Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 3
      Number of SNES iterations = 3

  Note this requires many fewer iterations than the default solver but requires more linear solver iterations than geometric multigrid.

- Run with the same options but use the ML preconditioner from Trilinos

      $ mpiexec -n 4 ./ex19 -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type ml

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
        0 SNES Function norm 1.036007954337e-02
          0 KSP Residual norm 4.888657920112e-01
          1 KSP Residual norm 3.141627616286e-01
          2 KSP Residual norm 2.358812378525e-01
          3 KSP Residual norm 1.259045437573e-01
          4 KSP Residual norm 7.408575133258e-02
          5 KSP Residual norm 4.159658325629e-02
          6 KSP Residual norm 2.181861584091e-02
          7 KSP Residual norm 1.076244407256e-02
          8 KSP Residual norm 5.242924144023e-03
          9 KSP Residual norm 2.588726461570e-03
         10 KSP Residual norm 1.400700093556e-03
         11 KSP Residual norm 7.759158101290e-04
         12 KSP Residual norm 4.655744671687e-04
         13 KSP Residual norm 3.244240900196e-04
         14 KSP Residual norm 2.327144448481e-04
         15 KSP Residual norm 1.500951563875e-04
         16 KSP Residual norm 8.327955132177e-05
         17 KSP Residual norm 4.672468799190e-05
         18 KSP Residual norm 2.889321996959e-05
         19 KSP Residual norm 1.903514112532e-05
         20 KSP Residual norm 1.298123631775e-05
         21 KSP Residual norm 8.775705379722e-06
         22 KSP Residual norm 5.330175482523e-06
         23 KSP Residual norm 2.869775929854e-06
        1 SNES Function norm 1.370527153606e-06
          0 KSP Residual norm 1.177060567864e-05
          1 KSP Residual norm 6.928128579778e-06
          2 KSP Residual norm 4.304601930959e-06
          3 KSP Residual norm 2.454875323346e-06
          4 KSP Residual norm 1.512790232898e-06
          5 KSP Residual norm 1.001915525185e-06
          6 KSP Residual norm 7.387623107528e-07
          7 KSP Residual norm 4.953711915641e-07
          8 KSP Residual norm 3.144821641519e-07
          9 KSP Residual norm 2.134819078071e-07
         10 KSP Residual norm 1.518551249827e-07
         11 KSP Residual norm 1.014185303569e-07
         12 KSP Residual norm 5.608673008996e-08
         13 KSP Residual norm 3.030519067648e-08
         14 KSP Residual norm 1.786947998191e-08
         15 KSP Residual norm 1.177816744649e-08
         16 KSP Residual norm 8.997335866987e-09
         17 KSP Residual norm 6.356106556680e-09
         18 KSP Residual norm 4.147037399694e-09
         19 KSP Residual norm 2.711154831563e-09
         20 KSP Residual norm 1.750865415567e-09
         21 KSP Residual norm 1.127681613823e-09
         22 KSP Residual norm 7.783034009647e-10
         23 KSP Residual norm 5.047971816405e-10
         24 KSP Residual norm 3.137090395920e-10
         25 KSP Residual norm 2.182629803057e-10
         26 KSP Residual norm 1.265836293994e-10
         27 KSP Residual norm 7.756916170515e-11
        2 SNES Function norm 2.339734806142e-11
      SNES Object: 4 MPI processes
        type: newtonls
        maximum iterations=50, maximum function evaluations=10000
        tolerances: relative=1e-08, absolute=1e-50, solution=1e-08
        total number of linear solver iterations=50
        total number of function evaluations=3
        norm schedule ALWAYS
        Jacobian is built using colored finite differences on a DMDA
        SNESLineSearch Object: 4 MPI processes
          type: bt
            interpolation: cubic
            alpha=1.000000e-04
          maxlambda=1.000000e+00, minlambda=1.000000e-12
          tolerances: relative=1.000000e-08, absolute=1.000000e-15, lambda=1.000000e-08
          maximum iterations=40
        KSP Object: 4 MPI processes
          type: gmres
            restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
            happy breakdown tolerance 1e-30
          maximum iterations=10000, initial guess is zero
          tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
          left preconditioning
          using PRECONDITIONED norm type for convergence test
        PC Object: 4 MPI processes
          type: ml
            type is MULTIPLICATIVE, levels=5 cycles=v
              Cycles per PCApply=1
              Using externally compute Galerkin coarse grid matrices
          Coarse grid solver -- level 0 -------------------------------
            KSP Object: (mg_coarse_) 4 MPI processes
              type: preonly
              maximum iterations=10000, initial guess is zero
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_coarse_) 4 MPI processes
              type: redundant
                First (color=0) of 4 PCs follows
                KSP Object: (mg_coarse_redundant_) 1 MPI process
                  type: preonly
                  maximum iterations=10000, initial guess is zero
                  tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
                  left preconditioning
                  using NONE norm type for convergence test
                PC Object: (mg_coarse_redundant_) 1 MPI process
                  type: lu
                    out-of-place factorization
                    tolerance for zero pivot 2.22045e-14
                    using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
                    matrix ordering: nd
                    factor fill ratio given 5., needed 1.
                      Factored matrix follows:
                        Mat Object: (mg_coarse_redundant_) 1 MPI process
                          type: seqaij
                          rows=16, cols=16, bs=4
                          package used to perform factorization: petsc
                          total: nonzeros=256, allocated nonzeros=256
                            using I-node routines: found 4 nodes, limit used is 5
                  linear system matrix = precond matrix:
                  Mat Object: 1 MPI process
                    type: seqaij
                    rows=16, cols=16, bs=4
                    total: nonzeros=256, allocated nonzeros=256
                    total number of mallocs used during MatSetValues calls=0
                      using I-node routines: found 4 nodes, limit used is 5
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=16, cols=16, bs=4
                total: nonzeros=256, allocated nonzeros=256
                total number of mallocs used during MatSetValues calls=0
                  using I-node (on process 0) routines: found 1 nodes, limit used is 5
          Down solver (pre-smoother) on level 1 -------------------------------
            KSP Object: (mg_levels_1_) 4 MPI processes
              type: richardson
                damping factor=1.
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_1_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=88, cols=88, bs=4
                total: nonzeros=4960, allocated nonzeros=4960
                total number of mallocs used during MatSetValues calls=0
                  using I-node (on process 0) routines: found 4 nodes, limit used is 5
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 2 -------------------------------
            KSP Object: (mg_levels_2_) 4 MPI processes
              type: richardson
                damping factor=1.
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_2_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=764, cols=764, bs=4
                total: nonzeros=35672, allocated nonzeros=35672
                total number of mallocs used during MatSetValues calls=0
                  using I-node (on process 0) routines: found 50 nodes, limit used is 5
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 3 -------------------------------
            KSP Object: (mg_levels_3_) 4 MPI processes
              type: richardson
                damping factor=1.
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_3_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=7156, cols=7156, bs=4
                total: nonzeros=259236, allocated nonzeros=259236
                total number of mallocs used during MatSetValues calls=0
                  not using I-node (on process 0) routines
          Up solver (post-smoother) same as down solver (pre-smoother)
          Down solver (pre-smoother) on level 4 -------------------------------
            KSP Object: (mg_levels_4_) 4 MPI processes
              type: richardson
                damping factor=1.
              maximum iterations=2, nonzero initial guess
              tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
              left preconditioning
              using NONE norm type for convergence test
            PC Object: (mg_levels_4_) 4 MPI processes
              type: sor
                type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
              linear system matrix = precond matrix:
              Mat Object: 4 MPI processes
                type: mpiaij
                rows=37636, cols=37636, bs=4
                total: nonzeros=746512, allocated nonzeros=746512
                total number of mallocs used during MatSetValues calls=0
          Up solver (post-smoother) same as down solver (pre-smoother)
          linear system matrix = precond matrix:
          Mat Object: 4 MPI processes
            type: mpiaij
            rows=37636, cols=37636, bs=4
            total: nonzeros=746512, allocated nonzeros=746512
            total number of mallocs used during MatSetValues calls=0
      Number of SNES iterations = 2

- Run on 1 processor with the default linear solver and profile the run

      $ mpiexec -n 1 ./ex19 -da_refine 5 -log_view

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
      Number of SNES iterations = 2
      ************************************************************************************************************************
      ***             WIDEN YOUR WINDOW TO 120 CHARACTERS.  Use 'enscript -r -fCourier9' to print this document            ***
      ************************************************************************************************************************

      ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------

      ./ex19 on a arch-ubuntu-mb-double-extra-opt named pdsbox with 1 processor, by patrick Fri Jul 27 15:30:21 2018
      Using Petsc Development GIT revision: v3.9.3-921-gfc2aa81  GIT Date: 2018-07-27 11:07:58 +0200

                               Max       Max/Min     Avg       Total
      Time (sec):           3.068e+00     1.000   3.068e+00
      Objects:              9.400e+01     1.000   9.400e+01
      Flops:                3.195e+09     1.000   3.195e+09  3.195e+09
      Flops/sec:            1.041e+09     1.000   1.041e+09  1.041e+09
      MPI Msg Count:        0.000e+00     0.000   0.000e+00  0.000e+00
      MPI Msg Len (bytes):  0.000e+00     0.000   0.000e+00  0.000e+00
      MPI Reductions:       0.000e+00     0.000

      Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
                                  e.g., VecAXPY() for real vectors of length N --> 2N flops
                                  and VecAXPY() for complex vectors of length N --> 8N flops

      Summary of Stages:   ----- Time ------  ----- Flop ------  --- Messages ---  -- Message Lengths --  -- Reductions --
                              Avg     %Total     Avg     %Total    Count   %Total     Avg         %Total    Count   %Total
       0:      Main Stage: 3.0680e+00 100.0%  3.1952e+09 100.0%  0.000e+00   0.0%  0.000e+00        0.0%  0.000e+00   0.0%

      ------------------------------------------------------------------------------------------------------------------------
      See the 'Profiling' chapter of the users' manual for details on interpreting output.
      Phase summary info:
         Count: number of times phase was executed
         Time and Flop: Max - maximum over all processors
                        Ratio - ratio of maximum to minimum over all processors
         Mess: number of messages sent
         AvgLen: average message length (bytes)
         Reduct: number of global reductions
         Global: entire computation
         Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
            %T - percent time in this phase         %F - percent flop in this phase
            %M - percent messages in this phase     %L - percent message lengths in this phase
            %R - percent reductions in this phase
         Total Mflop/s: 1e-6 * (sum of flop over all processors)/(max time over all processors)
      ------------------------------------------------------------------------------------------------------------------------
      Event                Count      Time (sec)     Flop                              --- Global ---  --- Stage ----  Total
                         Max Ratio  Max     Ratio   Max  Ratio  Mess   AvgLen  Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
      ------------------------------------------------------------------------------------------------------------------------

      --- Event Stage 0: Main Stage

      SNESSolve              1 1.0 3.0142e+00 1.0 3.20e+09 1.0 0.0e+00 0.0e+00 0.0e+00 98100  0  0  0  98100  0  0  0  1060
      SNESFunctionEval      45 1.0 4.6601e-02 1.0 3.56e+07 1.0 0.0e+00 0.0e+00 0.0e+00  2  1  0  0  0   2  1  0  0  0   763
      SNESJacobianEval       2 1.0 8.7663e-02 1.0 3.63e+07 1.0 0.0e+00 0.0e+00 0.0e+00  3  1  0  0  0   3  1  0  0  0   414
      SNESLineSearch         2 1.0 6.2666e-03 1.0 5.32e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   849
      VecDot                 2 1.0 7.8201e-05 1.0 1.51e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1925
      VecMDot              568 1.0 2.6999e-01 1.0 6.61e+08 1.0 0.0e+00 0.0e+00 0.0e+00  9 21  0  0  0   9 21  0  0  0  2447
      VecNorm              593 1.0 1.2326e-02 1.0 4.46e+07 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  3621
      VecScale             587 1.0 7.1690e-03 1.0 2.21e+07 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  3082
      VecCopy               63 1.0 1.7498e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecSet                64 1.0 4.0205e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecAXPY               76 1.0 2.5930e-03 1.0 5.72e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  2206
      VecWAXPY               2 1.0 2.6870e-04 1.0 7.53e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   280
      VecMAXPY             587 1.0 4.1431e-01 1.0 7.03e+08 1.0 0.0e+00 0.0e+00 0.0e+00 14 22  0  0  0  14 22  0  0  0  1698
      VecScatterBegin       45 1.0 7.8702e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecReduceArith         4 1.0 9.0122e-05 1.0 3.01e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  3341
      VecReduceComm          2 1.0 1.6689e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecNormalize         587 1.0 2.0009e-02 1.0 6.63e+07 1.0 0.0e+00 0.0e+00 0.0e+00  1  2  0  0  0   1  2  0  0  0  3312
      MatMult              587 1.0 8.6304e-01 1.0 8.54e+08 1.0 0.0e+00 0.0e+00 0.0e+00 28 27  0  0  0  28 27  0  0  0   990
      MatSolve             587 1.0 1.2751e+00 1.0 8.54e+08 1.0 0.0e+00 0.0e+00 0.0e+00 42 27  0  0  0  42 27  0  0  0   670
      MatLUFactorNum         2 1.0 5.5508e-02 1.0 1.41e+07 1.0 0.0e+00 0.0e+00 0.0e+00  2  0  0  0  0   2  0  0  0  0   254
      MatILUFactorSym        1 1.0 9.9506e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatAssemblyBegin       3 1.0 1.4305e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatAssemblyEnd         3 1.0 6.2709e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatGetRowIJ            1 1.0 4.0531e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatGetOrdering         1 1.0 1.2467e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatZeroEntries         2 1.0 4.0841e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorCreate       1 1.0 2.3603e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorSetUp        1 1.0 2.7331e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      MatFDColorApply        2 1.0 5.9345e-02 1.0 3.63e+07 1.0 0.0e+00 0.0e+00 0.0e+00  2  1  0  0  0   2  1  0  0  0   611
      MatFDColorFunc        42 1.0 4.4162e-02 1.0 3.32e+07 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0   752
      KSPSetUp               2 1.0 6.0439e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      KSPSolve               2 1.0 2.9187e+00 1.0 3.15e+09 1.0 0.0e+00 0.0e+00 0.0e+00 95 99  0  0  0  95 99  0  0  0  1080
      KSPGMRESOrthog       568 1.0 6.6268e-01 1.0 1.32e+09 1.0 0.0e+00 0.0e+00 0.0e+00 22 41  0  0  0  22 41  0  0  0  1994
      PCSetUp                2 1.0 6.6768e-02 1.0 1.41e+07 1.0 0.0e+00 0.0e+00 0.0e+00  2  0  0  0  0   2  0  0  0  0   211
      PCApply              587 1.0 1.2759e+00 1.0 8.54e+08 1.0 0.0e+00 0.0e+00 0.0e+00 42 27  0  0  0  42 27  0  0  0   670
      ------------------------------------------------------------------------------------------------------------------------

      Memory usage is given in bytes:

      Object Type          Creations   Destructions     Memory  Descendants' Mem.
      Reports information only for process 0.

      --- Event Stage 0: Main Stage

                      SNES     1              1         1372     0.
                    DMSNES     1              1          672     0.
            SNESLineSearch     1              1         1000     0.
                    Vector    47             47     13320304     0.
                    Matrix     2              2     19277596     0.
        Matrix FD Coloring     1              1     16612048     0.
          Distributed Mesh     1              1         5344     0.
                 Index Set    29             29       738052     0.
         IS L to G Mapping     2              2       189524     0.
         Star Forest Graph     2              2         1728     0.
           Discrete System     1              1          932     0.
               Vec Scatter     2              2         1408     0.
             Krylov Solver     1              1        18416     0.
           DMKSP interface     1              1          656     0.
            Preconditioner     1              1         1008     0.
                    Viewer     1              0            0     0.
      ========================================================================================================================
      Average time to get PetscTime(): 4.76837e-08
      #PETSc Option Table entries:
      -da_refine 5
      -log_view
      #End of PETSc Option Table entries
      Compiled without FORTRAN kernels
      Compiled with full precision matrices (default)
      sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
      Configure options: PETSC_DIR=/home/patrick/petsc-mb PETSC_ARCH=arch-ubuntu-mb-double-extra-opt --with-debugging=0 --with-precision=double --with-scalar-type=real --with-cc=gcc --with-cxx=g++ --with-fc=gfortran --download-mpich --COPTFLAGS="-g -O3 -march=native " --CXXOPTFLAGS="-g -O3 -march=native " --FOPTFLAGS="-g -O3 -march=native " --download-c2html --download-suitesparse --download-yaml --download-hdf5 --download-scalapack --download-metis --download-parmetis --download-mumps --download-superlu_dist --download-triangle --download-ctetgen --download-sundials --download-ml --download-exodusii --download-hdf5 --download-netcdf --download-pnetcdf
      -----------------------------------------
      Libraries compiled on 2018-07-27 13:01:14 on pdsbox
      Machine characteristics: Linux-4.13.0-39-generic-x86_64-with-Ubuntu-16.04-xenial
      Using PETSc directory: /home/patrick/petsc-mb
      Using PETSc arch: arch-ubuntu-mb-double-extra-opt
      -----------------------------------------

      Using C compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc    -Wall -Wwrite-strings -Wno-strict-aliasing -Wno-unknown-pragmas -fstack-protector -fvisibility=hidden
      Using Fortran compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90   -Wall -ffree-line-length-0 -Wno-unused-dummy-argument
      -----------------------------------------

      Using include paths: -I/home/patrick/petsc-mb/include -I/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/include
      -----------------------------------------

      Using C linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc
      Using Fortran linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90
      Using libraries: -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -lpetsc -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -Wl,-rpath,/usr/lib/gcc/x86_64-linux-gnu/5 -L/usr/lib/gcc/x86_64-linux-gnu/5 -Wl,-rpath,/usr/lib/x86_64-linux-gnu -L/usr/lib/x86_64-linux-gnu -Wl,-rpath,/lib/x86_64-linux-gnu -L/lib/x86_64-linux-gnu -lcmumps -ldmumps -lsmumps -lzmumps -lmumps_common -lpord -lpthread -lscalapack -lumfpack -lklu -lcholmod -lbtf -lccolamd -lcolamd -lcamd -lamd -lsuitesparseconfig -lsuperlu_dist -lml -lsundials_cvode -lsundials_nvecserial -lsundials_nvecparallel -llapack -lblas -lexodus -lnetcdf -lpnetcdf -lhdf5hl_fortran -lhdf5_fortran -lhdf5_hl -lhdf5 -lparmetis -lmetis -ltriangle -lm -lctetgen -lpthread -lyaml -lstdc++ -ldl -lmpifort -lmpi -lgfortran -lm -lgfortran -lm -lgcc_s -lquadmath -lstdc++ -ldl
      -----------------------------------------

  Search for the line beginning with SNESSolve, the fourth column gives the time for the nonlinear solve.

- Run on 1 processor with the geometric multigrid linear solver and profile the run

      $ mpiexec -n 1 ./ex19 -da_refine 5 -log_view -pc_type mg

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
      Number of SNES iterations = 3
      ************************************************************************************************************************
      ***             WIDEN YOUR WINDOW TO 120 CHARACTERS.  Use 'enscript -r -fCourier9' to print this document            ***
      ************************************************************************************************************************

      ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------

      ./ex19 on a arch-ubuntu-mb-double-extra-opt named pdsbox with 1 processor, by patrick Fri Jul 27 15:30:40 2018
      Using Petsc Development GIT revision: v3.9.3-921-gfc2aa81  GIT Date: 2018-07-27 11:07:58 +0200

                               Max       Max/Min     Avg       Total
      Time (sec):           6.992e-01     1.000   6.992e-01
      Objects:              4.800e+02     1.000   4.800e+02
      Flops:                5.237e+08     1.000   5.237e+08  5.237e+08
      Flops/sec:            7.490e+08     1.000   7.490e+08  7.490e+08
      MPI Msg Count:        0.000e+00     0.000   0.000e+00  0.000e+00
      MPI Msg Len (bytes):  0.000e+00     0.000   0.000e+00  0.000e+00
      MPI Reductions:       0.000e+00     0.000

      Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
                                  e.g., VecAXPY() for real vectors of length N --> 2N flops
                                  and VecAXPY() for complex vectors of length N --> 8N flops

      Summary of Stages:   ----- Time ------  ----- Flop ------  --- Messages ---  -- Message Lengths --  -- Reductions --
                              Avg     %Total     Avg     %Total    Count   %Total     Avg         %Total    Count   %Total
       0:      Main Stage: 6.9923e-01 100.0%  5.2371e+08 100.0%  0.000e+00   0.0%  0.000e+00        0.0%  0.000e+00   0.0%

      ------------------------------------------------------------------------------------------------------------------------
      See the 'Profiling' chapter of the users' manual for details on interpreting output.
      Phase summary info:
         Count: number of times phase was executed
         Time and Flop: Max - maximum over all processors
                        Ratio - ratio of maximum to minimum over all processors
         Mess: number of messages sent
         AvgLen: average message length (bytes)
         Reduct: number of global reductions
         Global: entire computation
         Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
            %T - percent time in this phase         %F - percent flop in this phase
            %M - percent messages in this phase     %L - percent message lengths in this phase
            %R - percent reductions in this phase
         Total Mflop/s: 1e-6 * (sum of flop over all processors)/(max time over all processors)
      ------------------------------------------------------------------------------------------------------------------------
      Event                Count      Time (sec)     Flop                              --- Global ---  --- Stage ----  Total
                         Max Ratio  Max     Ratio   Max  Ratio  Mess   AvgLen  Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
      ------------------------------------------------------------------------------------------------------------------------

      --- Event Stage 0: Main Stage

      SNESSolve              1 1.0 6.4513e-01 1.0 5.24e+08 1.0 0.0e+00 0.0e+00 0.0e+00 92100  0  0  0  92100  0  0  0   812
      SNESFunctionEval     255 1.0 5.6777e-02 1.0 4.71e+07 1.0 0.0e+00 0.0e+00 0.0e+00  8  9  0  0  0   8  9  0  0  0   829
      SNESJacobianEval      12 1.0 1.1569e-01 1.0 4.89e+07 1.0 0.0e+00 0.0e+00 0.0e+00 17  9  0  0  0  17  9  0  0  0   422
      SNESLineSearch         2 1.0 5.7764e-03 1.0 5.32e+06 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0   921
      VecDot                 2 1.0 7.6056e-05 1.0 1.51e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1979
      VecMDot              111 1.0 7.1726e-03 1.0 1.38e+07 1.0 0.0e+00 0.0e+00 0.0e+00  1  3  0  0  0   1  3  0  0  0  1930
      VecNorm              139 1.0 1.0304e-03 1.0 3.71e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  3600
      VecScale             123 1.0 6.1989e-04 1.0 1.60e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  2586
      VecCopy              321 1.0 1.7195e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecSet               382 1.0 4.5128e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      VecAXPY              252 1.0 3.0298e-03 1.0 4.41e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  1455
      VecAYPX              520 1.0 1.4517e-02 1.0 6.58e+06 1.0 0.0e+00 0.0e+00 0.0e+00  2  1  0  0  0   2  1  0  0  0   453
      VecAXPBYCZ           260 1.0 9.2232e-03 1.0 1.32e+07 1.0 0.0e+00 0.0e+00 0.0e+00  1  3  0  0  0   1  3  0  0  0  1427
      VecWAXPY               2 1.0 2.7275e-04 1.0 7.53e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   276
      VecMAXPY             123 1.0 1.3335e-02 1.0 1.67e+07 1.0 0.0e+00 0.0e+00 0.0e+00  2  3  0  0  0   2  3  0  0  0  1252
      VecScatterBegin      265 1.0 1.1539e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecReduceArith         4 1.0 8.3208e-05 1.0 3.01e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  3618
      VecReduceComm          2 1.0 1.4305e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecNormalize         123 1.0 1.6305e-03 1.0 4.81e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  2949
      MatMult              513 1.0 1.9088e-01 1.0 2.10e+08 1.0 0.0e+00 0.0e+00 0.0e+00 27 40  0  0  0  27 40  0  0  0  1100
      MatMultAdd            65 1.0 5.0337e-03 1.0 2.93e+06 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0   582
      MatMultTranspose      70 1.0 5.1179e-03 1.0 3.16e+06 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0   617
      MatSolve              13 1.0 1.2708e-04 1.0 4.91e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   386
      MatSOR               500 1.0 2.4601e-01 1.0 2.00e+08 1.0 0.0e+00 0.0e+00 0.0e+00 35 38  0  0  0  35 38  0  0  0   813
      MatLUFactorSym         1 1.0 1.1539e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatLUFactorNum         2 1.0 1.2755e-04 1.0 1.82e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   143
      MatResidual           65 1.0 2.3766e-02 1.0 2.60e+07 1.0 0.0e+00 0.0e+00 0.0e+00  3  5  0  0  0   3  5  0  0  0  1094
      MatAssemblyBegin      23 1.0 6.9141e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatAssemblyEnd        23 1.0 8.9450e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      MatGetRowIJ            1 1.0 1.2636e-05 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatGetOrdering         1 1.0 4.2200e-05 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatZeroEntries        12 1.0 5.6696e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorCreate       6 1.0 2.8205e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorSetUp        6 1.0 3.7257e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  5  0  0  0  0   5  0  0  0  0     0
      MatFDColorApply       12 1.0 7.7073e-02 1.0 4.89e+07 1.0 0.0e+00 0.0e+00 0.0e+00 11  9  0  0  0  11  9  0  0  0   634
      MatFDColorFunc       252 1.0 5.5366e-02 1.0 4.47e+07 1.0 0.0e+00 0.0e+00 0.0e+00  8  9  0  0  0   8  9  0  0  0   807
      DMCoarsen              5 1.0 2.9874e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      DMCreateInterp         5 1.0 3.7532e-03 1.0 2.25e+05 1.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0    60
      KSPSetUp              19 1.0 9.5367e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      KSPSolve               2 1.0 5.5405e-01 1.0 4.81e+08 1.0 0.0e+00 0.0e+00 0.0e+00 79 92  0  0  0  79 92  0  0  0   869
      KSPGMRESOrthog       111 1.0 1.8478e-02 1.0 2.77e+07 1.0 0.0e+00 0.0e+00 0.0e+00  3  5  0  0  0   3  5  0  0  0  1498
      PCSetUp                2 1.0 5.5341e-02 1.0 1.28e+07 1.0 0.0e+00 0.0e+00 0.0e+00  8  2  0  0  0   8  2  0  0  0   232
      PCApply               13 1.0 4.7885e-01 1.0 4.45e+08 1.0 0.0e+00 0.0e+00 0.0e+00 68 85  0  0  0  68 85  0  0  0   928
      ------------------------------------------------------------------------------------------------------------------------

      Memory usage is given in bytes:

      Object Type          Creations   Destructions     Memory  Descendants' Mem.
      Reports information only for process 0.

      --- Event Stage 0: Main Stage

                      SNES     1              1         1372     0.
                    DMSNES     6              6         4432     0.
            SNESLineSearch     1              1         1000     0.
                    Vector   206            206     18098080     0.
                    Matrix    22             22     13706952     0.
        Matrix FD Coloring     6              6     22297904     0.
          Distributed Mesh     6              6        31664     0.
                 Index Set   159            159       393244     0.
         IS L to G Mapping    12             12       261444     0.
         Star Forest Graph    12             12         9728     0.
           Discrete System     6              6         5572     0.
               Vec Scatter    17             17        11968     0.
             Krylov Solver    12             12       177272     0.
           DMKSP interface     6              6         3936     0.
            Preconditioner     7              7         6968     0.
                    Viewer     1              0            0     0.
      ========================================================================================================================
      Average time to get PetscTime(): 4.76837e-08
      #PETSc Option Table entries:
      -da_refine 5
      -log_view
      -pc_type mg
      #End of PETSc Option Table entries
      Compiled without FORTRAN kernels
      Compiled with full precision matrices (default)
      sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
      Configure options: PETSC_DIR=/home/patrick/petsc-mb PETSC_ARCH=arch-ubuntu-mb-double-extra-opt --with-debugging=0 --with-precision=double --with-scalar-type=real --with-cc=gcc --with-cxx=g++ --with-fc=gfortran --download-mpich --COPTFLAGS="-g -O3 -march=native " --CXXOPTFLAGS="-g -O3 -march=native " --FOPTFLAGS="-g -O3 -march=native " --download-c2html --download-suitesparse --download-yaml --download-hdf5 --download-scalapack --download-metis --download-parmetis --download-mumps --download-superlu_dist --download-triangle --download-ctetgen --download-sundials --download-ml --download-exodusii --download-hdf5 --download-netcdf --download-pnetcdf
      -----------------------------------------
      Libraries compiled on 2018-07-27 13:01:14 on pdsbox
      Machine characteristics: Linux-4.13.0-39-generic-x86_64-with-Ubuntu-16.04-xenial
      Using PETSc directory: /home/patrick/petsc-mb
      Using PETSc arch: arch-ubuntu-mb-double-extra-opt
      -----------------------------------------

      Using C compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc    -Wall -Wwrite-strings -Wno-strict-aliasing -Wno-unknown-pragmas -fstack-protector -fvisibility=hidden
      Using Fortran compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90   -Wall -ffree-line-length-0 -Wno-unused-dummy-argument
      -----------------------------------------

      Using include paths: -I/home/patrick/petsc-mb/include -I/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/include
      -----------------------------------------

      Using C linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc
      Using Fortran linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90
      Using libraries: -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -lpetsc -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -Wl,-rpath,/usr/lib/gcc/x86_64-linux-gnu/5 -L/usr/lib/gcc/x86_64-linux-gnu/5 -Wl,-rpath,/usr/lib/x86_64-linux-gnu -L/usr/lib/x86_64-linux-gnu -Wl,-rpath,/lib/x86_64-linux-gnu -L/lib/x86_64-linux-gnu -lcmumps -ldmumps -lsmumps -lzmumps -lmumps_common -lpord -lpthread -lscalapack -lumfpack -lklu -lcholmod -lbtf -lccolamd -lcolamd -lcamd -lamd -lsuitesparseconfig -lsuperlu_dist -lml -lsundials_cvode -lsundials_nvecserial -lsundials_nvecparallel -llapack -lblas -lexodus -lnetcdf -lpnetcdf -lhdf5hl_fortran -lhdf5_fortran -lhdf5_hl -lhdf5 -lparmetis -lmetis -ltriangle -lm -lctetgen -lpthread -lyaml -lstdc++ -ldl -lmpifort -lmpi -lgfortran -lm -lgfortran -lm -lgcc_s -lquadmath -lstdc++ -ldl
      -----------------------------------------

  Compare the runtime for SNESSolve to the case with the default solver

- Run on 4 processors with the default linear solver and profile the run

      $ mpiexec -n 4 ./ex19 -da_refine 5 -log_view

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
      Number of SNES iterations = 2
      ************************************************************************************************************************
      ***             WIDEN YOUR WINDOW TO 120 CHARACTERS.  Use 'enscript -r -fCourier9' to print this document            ***
      ************************************************************************************************************************

      ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------

      ./ex19 on a arch-ubuntu-mb-double-extra-opt named pdsbox with 4 processors, by patrick Fri Jul 27 15:30:00 2018
      Using Petsc Development GIT revision: v3.9.3-921-gfc2aa81  GIT Date: 2018-07-27 11:07:58 +0200

                               Max       Max/Min     Avg       Total
      Time (sec):           1.200e+00     1.000   1.200e+00
      Objects:              9.900e+01     1.000   9.900e+01
      Flops:                9.349e+08     1.042   9.158e+08  3.663e+09
      Flops/sec:            7.789e+08     1.042   7.631e+08  3.053e+09
      MPI Msg Count:        1.453e+03     1.000   1.453e+03  5.811e+03
      MPI Msg Len (bytes):  2.266e+06     1.021   1.544e+03  8.972e+06
      MPI Reductions:       1.535e+03     1.000

      Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
                                  e.g., VecAXPY() for real vectors of length N --> 2N flops
                                  and VecAXPY() for complex vectors of length N --> 8N flops

      Summary of Stages:   ----- Time ------  ----- Flop ------  --- Messages ---  -- Message Lengths --  -- Reductions --
                              Avg     %Total     Avg     %Total    Count   %Total     Avg         %Total    Count   %Total
       0:      Main Stage: 1.2001e+00 100.0%  3.6633e+09 100.0%  5.811e+03 100.0%  1.544e+03      100.0%  1.528e+03  99.5%

      ------------------------------------------------------------------------------------------------------------------------
      See the 'Profiling' chapter of the users' manual for details on interpreting output.
      Phase summary info:
         Count: number of times phase was executed
         Time and Flop: Max - maximum over all processors
                        Ratio - ratio of maximum to minimum over all processors
         Mess: number of messages sent
         AvgLen: average message length (bytes)
         Reduct: number of global reductions
         Global: entire computation
         Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
            %T - percent time in this phase         %F - percent flop in this phase
            %M - percent messages in this phase     %L - percent message lengths in this phase
            %R - percent reductions in this phase
         Total Mflop/s: 1e-6 * (sum of flop over all processors)/(max time over all processors)
      ------------------------------------------------------------------------------------------------------------------------
      Event                Count      Time (sec)     Flop                              --- Global ---  --- Stage ----  Total
                         Max Ratio  Max     Ratio   Max  Ratio  Mess   AvgLen  Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
      ------------------------------------------------------------------------------------------------------------------------

      --- Event Stage 0: Main Stage

      BuildTwoSidedF         3 1.0 5.1808e-0314.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      SNESSolve              1 1.0 1.1392e+00 1.0 9.35e+08 1.0 5.8e+03 1.6e+03 1.5e+03 95100 99100 98  95100 99100 98  3216
      SNESFunctionEval      45 1.0 6.7165e-03 1.0 9.08e+06 1.0 2.4e+01 1.6e+03 0.0e+00  1  1  0  0  0   1  1  0  0  0  5295
      SNESJacobianEval       2 1.0 2.6000e-02 1.0 9.26e+06 1.0 3.4e+02 1.5e+03 8.6e+01  2  1  6  6  6   2  1  6  6  6  1395
      SNESLineSearch         2 1.0 1.9200e-03 1.0 1.36e+06 1.0 3.2e+01 1.6e+03 8.0e+00  0  0  1  1  1   0  0  1  1  1  2771
      VecDot                 2 1.0 2.2244e-04 2.2 3.84e+04 1.0 0.0e+00 0.0e+00 2.0e+00  0  0  0  0  0   0  0  0  0  0   677
      VecMDot              655 1.0 2.0498e-01 1.1 1.94e+08 1.0 0.0e+00 0.0e+00 6.6e+02 16 21  0  0 43  16 21  0  0 43  3705
      VecNorm              683 1.0 4.9419e-02 1.3 1.31e+07 1.0 0.0e+00 0.0e+00 6.8e+02  4  1  0  0 44   4  1  0  0 45  1040
      VecScale             677 1.0 2.5268e-03 1.1 6.50e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0 10084
      VecCopy               66 1.0 5.6410e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecSet               703 1.0 2.7184e-03 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecAXPY               82 1.0 5.7197e-04 1.1 1.58e+06 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0 10791
      VecWAXPY               2 1.0 5.7220e-05 1.1 1.92e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1315
      VecMAXPY             677 1.0 1.2249e-01 1.1 2.06e+08 1.0 0.0e+00 0.0e+00 0.0e+00 10 22  0  0  0  10 22  0  0  0  6603
      VecScatterBegin      722 1.0 9.7113e-03 1.1 0.00e+00 0.0 5.8e+03 1.6e+03 0.0e+00  1  0 99100  0   1  0 99100  0     0
      VecScatterEnd        722 1.0 1.0391e-02 1.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      VecReduceArith         4 1.0 4.0293e-05 1.1 7.68e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  7472
      VecReduceComm          2 1.0 7.3195e-05 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 2.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecNormalize         677 1.0 5.1487e-02 1.2 1.95e+07 1.0 0.0e+00 0.0e+00 6.8e+02  4  2  0  0 44   4  2  0  0 44  1485
      MatMult              677 1.0 3.2710e-01 1.0 2.51e+08 1.0 5.4e+03 1.6e+03 0.0e+00 27 27 93 94  0  27 27 93 94  0  3012
      MatSolve             677 1.0 3.9744e-01 1.0 2.49e+08 1.0 0.0e+00 0.0e+00 0.0e+00 33 27  0  0  0  33 27  0  0  0  2458
      MatLUFactorNum         2 1.0 9.7592e-03 1.0 3.53e+06 1.1 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0  1405
      MatILUFactorSym        1 1.0 2.1026e-03 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatAssemblyBegin       3 1.0 5.2419e-0312.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatAssemblyEnd         3 1.0 3.2048e-03 1.0 0.00e+00 0.0 1.6e+01 3.9e+02 8.0e+00  0  0  0  0  1   0  0  0  0  1     0
      MatGetRowIJ            1 1.0 3.8147e-06 1.3 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatGetOrdering         1 1.0 2.5654e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatZeroEntries         2 1.0 1.5545e-04 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorCreate       1 1.0 1.1539e-04 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 2.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorSetUp        1 1.0 1.1923e-02 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 8.0e+01  1  0  0  0  5   1  0  0  0  5     0
      MatFDColorApply        2 1.0 1.4742e-02 1.1 9.26e+06 1.0 3.4e+02 1.6e+03 3.0e+00  1  1  6  6  0   1  1  6  6  0  2461
      MatFDColorFunc        42 1.0 8.3950e-03 1.2 8.47e+06 1.0 3.4e+02 1.6e+03 0.0e+00  1  1  6  6  0   1  1  6  6  0  3954
      KSPSetUp               4 1.0 4.4918e-04 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 1.0e+01  0  0  0  0  1   0  0  0  0  1     0
      KSPSolve               2 1.0 1.1108e+00 1.0 9.24e+08 1.0 5.4e+03 1.6e+03 1.4e+03 93 99 93 93 91  93 99 93 93 92  3260
      KSPGMRESOrthog       655 1.0 3.2223e-01 1.1 3.88e+08 1.0 0.0e+00 0.0e+00 6.6e+02 26 41  0  0 43  26 41  0  0 43  4714
      PCSetUp                4 1.0 1.2222e-02 1.0 3.53e+06 1.1 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0  1122
      PCSetUpOnBlocks        2 1.0 1.2162e-02 1.0 3.53e+06 1.1 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0  1128
      PCApply              677 1.0 4.1153e-01 1.0 2.49e+08 1.0 0.0e+00 0.0e+00 0.0e+00 34 27  0  0  0  34 27  0  0  0  2374
      ------------------------------------------------------------------------------------------------------------------------

      Memory usage is given in bytes:

      Object Type          Creations   Destructions     Memory  Descendants' Mem.
      Reports information only for process 0.

      --- Event Stage 0: Main Stage

                      SNES     1              1         1372     0.
                    DMSNES     1              1          672     0.
            SNESLineSearch     1              1         1000     0.
                    Vector    49             49      3390624     0.
                    Matrix     4              4      5105468     0.
        Matrix FD Coloring     1              1      4240080     0.
          Distributed Mesh     1              1         5344     0.
                 Index Set    29             29       207404     0.
         IS L to G Mapping     1              1        10672     0.
         Star Forest Graph     2              2         1728     0.
           Discrete System     1              1          932     0.
               Vec Scatter     2              2        22184     0.
             Krylov Solver     2              2        19592     0.
           DMKSP interface     1              1          656     0.
            Preconditioner     2              2         1912     0.
                    Viewer     1              0            0     0.
      ========================================================================================================================
      Average time to get PetscTime(): 2.38419e-08
      Average time for MPI_Barrier(): 2.91824e-05
      Average time for zero size MPI_Send(): 8.88109e-06
      #PETSc Option Table entries:
      -da_refine 5
      -log_view
      #End of PETSc Option Table entries
      Compiled without FORTRAN kernels
      Compiled with full precision matrices (default)
      sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
      Configure options: PETSC_DIR=/home/patrick/petsc-mb PETSC_ARCH=arch-ubuntu-mb-double-extra-opt --with-debugging=0 --with-precision=double --with-scalar-type=real --with-cc=gcc --with-cxx=g++ --with-fc=gfortran --download-mpich --COPTFLAGS="-g -O3 -march=native " --CXXOPTFLAGS="-g -O3 -march=native " --FOPTFLAGS="-g -O3 -march=native " --download-c2html --download-suitesparse --download-yaml --download-hdf5 --download-scalapack --download-metis --download-parmetis --download-mumps --download-superlu_dist --download-triangle --download-ctetgen --download-sundials --download-ml --download-exodusii --download-hdf5 --download-netcdf --download-pnetcdf
      -----------------------------------------
      Libraries compiled on 2018-07-27 13:01:14 on pdsbox
      Machine characteristics: Linux-4.13.0-39-generic-x86_64-with-Ubuntu-16.04-xenial
      Using PETSc directory: /home/patrick/petsc-mb
      Using PETSc arch: arch-ubuntu-mb-double-extra-opt
      -----------------------------------------

      Using C compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc    -Wall -Wwrite-strings -Wno-strict-aliasing -Wno-unknown-pragmas -fstack-protector -fvisibility=hidden
      Using Fortran compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90   -Wall -ffree-line-length-0 -Wno-unused-dummy-argument
      -----------------------------------------

      Using include paths: -I/home/patrick/petsc-mb/include -I/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/include
      -----------------------------------------

      Using C linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc
      Using Fortran linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90
      Using libraries: -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -lpetsc -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -Wl,-rpath,/usr/lib/gcc/x86_64-linux-gnu/5 -L/usr/lib/gcc/x86_64-linux-gnu/5 -Wl,-rpath,/usr/lib/x86_64-linux-gnu -L/usr/lib/x86_64-linux-gnu -Wl,-rpath,/lib/x86_64-linux-gnu -L/lib/x86_64-linux-gnu -lcmumps -ldmumps -lsmumps -lzmumps -lmumps_common -lpord -lpthread -lscalapack -lumfpack -lklu -lcholmod -lbtf -lccolamd -lcolamd -lcamd -lamd -lsuitesparseconfig -lsuperlu_dist -lml -lsundials_cvode -lsundials_nvecserial -lsundials_nvecparallel -llapack -lblas -lexodus -lnetcdf -lpnetcdf -lhdf5hl_fortran -lhdf5_fortran -lhdf5_hl -lhdf5 -lparmetis -lmetis -ltriangle -lm -lctetgen -lpthread -lyaml -lstdc++ -ldl -lmpifort -lmpi -lgfortran -lm -lgfortran -lm -lgcc_s -lquadmath -lstdc++ -ldl
      -----------------------------------------

  Compare the runtime for `SNESSolve` to the 1 processor case with the default solver. What is the speedup?

- Run on 4 processors with the geometric multigrid linear solver and profile the run

      $ mpiexec -n 4 ./ex19 -da_refine 5 -log_view -pc_type mg

  Expected output:

      lid velocity = 0.000106281, prandtl # = 1., grashof # = 1.
      Number of SNES iterations = 3
      ************************************************************************************************************************
      ***             WIDEN YOUR WINDOW TO 120 CHARACTERS.  Use 'enscript -r -fCourier9' to print this document            ***
      ************************************************************************************************************************

      ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------

      ./ex19 on a arch-ubuntu-mb-double-extra-opt named pdsbox with 4 processors, by patrick Fri Jul 27 15:29:45 2018
      Using Petsc Development GIT revision: v3.9.3-921-gfc2aa81  GIT Date: 2018-07-27 11:07:58 +0200

                               Max       Max/Min     Avg       Total
      Time (sec):           4.796e-01     1.001   4.794e-01
      Objects:              5.730e+02     1.000   5.730e+02
      Flops:                2.266e+08     1.061   2.201e+08  8.802e+08
      Flops/sec:            4.726e+08     1.060   4.590e+08  1.836e+09
      MPI Msg Count:        3.012e+03     1.043   2.950e+03  1.180e+04
      MPI Msg Len (bytes):  1.664e+06     1.054   5.490e+02  6.478e+06
      MPI Reductions:       1.472e+03     1.000

      Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
                                  e.g., VecAXPY() for real vectors of length N --> 2N flops
                                  and VecAXPY() for complex vectors of length N --> 8N flops

      Summary of Stages:   ----- Time ------  ----- Flop ------  --- Messages ---  -- Message Lengths --  -- Reductions --
                              Avg     %Total     Avg     %Total    Count   %Total     Avg         %Total    Count   %Total
       0:      Main Stage: 4.7942e-01 100.0%  8.8021e+08 100.0%  1.180e+04 100.0%  5.490e+02      100.0%  1.465e+03  99.5%

      ------------------------------------------------------------------------------------------------------------------------
      See the 'Profiling' chapter of the users' manual for details on interpreting output.
      Phase summary info:
         Count: number of times phase was executed
         Time and Flop: Max - maximum over all processors
                        Ratio - ratio of maximum to minimum over all processors
         Mess: number of messages sent
         AvgLen: average message length (bytes)
         Reduct: number of global reductions
         Global: entire computation
         Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
            %T - percent time in this phase         %F - percent flop in this phase
            %M - percent messages in this phase     %L - percent message lengths in this phase
            %R - percent reductions in this phase
         Total Mflop/s: 1e-6 * (sum of flop over all processors)/(max time over all processors)
      ------------------------------------------------------------------------------------------------------------------------
      Event                Count      Time (sec)     Flop                              --- Global ---  --- Stage ----  Total
                         Max Ratio  Max     Ratio   Max  Ratio  Mess   AvgLen  Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
      ------------------------------------------------------------------------------------------------------------------------

      --- Event Stage 0: Main Stage

      BuildTwoSidedF        29 1.0 5.8196e-03 6.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      SNESSolve              1 1.0 4.1890e-01 1.0 2.27e+08 1.1 1.2e+04 5.5e+02 1.4e+03 87100100100 98  87100100100 98  2101
      SNESFunctionEval     382 1.0 1.7268e-02 1.4 1.81e+07 1.1 3.2e+01 1.6e+03 0.0e+00  3  8  0  1  0   3  8  0  1  0  4066
      SNESJacobianEval      18 1.0 6.6740e-02 1.0 1.89e+07 1.1 3.0e+03 5.2e+02 5.3e+02 14  8 26 24 36  14  8 26 24 36  1098
      SNESLineSearch         3 1.0 2.8355e-03 1.0 2.04e+06 1.0 4.8e+01 1.6e+03 1.2e+01  1  1  0  1  1   1  1  0  1  1  2814
      VecDot                 3 1.0 2.3961e-04 1.3 5.76e+04 1.0 0.0e+00 0.0e+00 3.0e+00  0  0  0  0  0   0  0  0  0  0   942
      VecMDot              170 1.0 2.7825e-02 2.9 5.78e+06 1.1 0.0e+00 0.0e+00 1.7e+02  4  3  0  0 12   4  3  0  0 12   808
      VecNorm              211 1.0 1.4362e-02 1.5 1.48e+06 1.1 0.0e+00 0.0e+00 2.1e+02  3  1  0  0 14   3  1  0  0 14   401
      VecScale             188 1.0 4.2009e-04 1.2 6.51e+05 1.1 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  6036
      VecCopy              499 1.0 1.2398e-03 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecSet               504 1.0 4.2868e-04 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecAXPY              378 1.0 6.4588e-04 1.4 1.70e+06 1.1 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0 10235
      VecAYPX              920 1.0 1.0263e-02 1.4 3.00e+06 1.1 0.0e+00 0.0e+00 0.0e+00  2  1  0  0  0   2  1  0  0  0  1134
      VecAXPBYCZ           460 1.0 5.8384e-03 1.2 6.00e+06 1.1 0.0e+00 0.0e+00 0.0e+00  1  3  0  0  0   1  3  0  0  0  3988
      VecWAXPY               3 1.0 1.2279e-04 1.4 2.88e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   920
      VecMAXPY             188 1.0 7.0734e-03 1.4 6.95e+06 1.1 0.0e+00 0.0e+00 0.0e+00  1  3  0  0  0   1  3  0  0  0  3823
      VecScatterBegin     1541 1.0 1.5262e-02 1.1 0.00e+00 0.0 1.1e+04 5.6e+02 0.0e+00  3  0 97 98  0   3  0 97 98  0     0
      VecScatterEnd       1541 1.0 4.6712e-02 3.2 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  6  0  0  0  0   6  0  0  0  0     0
      VecReduceArith         6 1.0 6.8903e-05 1.3 1.15e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  6554
      VecReduceComm          3 1.0 2.2459e-04 1.6 0.00e+00 0.0 0.0e+00 0.0e+00 3.0e+00  0  0  0  0  0   0  0  0  0  0     0
      VecNormalize         188 1.0 1.3350e-02 1.6 1.95e+06 1.1 0.0e+00 0.0e+00 1.9e+02  2  1  0  0 13   2  1  0  0 13   570
      MatMult              878 1.0 1.2342e-01 1.2 9.29e+07 1.1 6.9e+03 6.4e+02 0.0e+00 23 41 59 68  0  23 41 59 68  0  2925
      MatMultAdd           115 1.0 4.8120e-03 1.1 1.32e+06 1.0 5.8e+02 2.6e+02 0.0e+00  1  1  5  2  0   1  1  5  2  0  1078
      MatMultTranspose     120 1.0 1.4319e-02 3.1 1.38e+06 1.0 6.0e+02 2.6e+02 0.0e+00  2  1  5  2  0   2  1  5  2  0   378
      MatSolve              23 1.0 2.2483e-04 1.1 8.68e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1545
      MatSOR               855 1.0 1.5684e-01 1.3 8.70e+07 1.1 0.0e+00 0.0e+00 0.0e+00 28 38  0  0  0  28 38  0  0  0  2154
      MatLUFactorSym         1 1.0 1.2636e-04 1.5 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatLUFactorNum         3 1.0 1.7381e-04 1.2 2.73e+04 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   628
      MatCopy                2 1.0 3.3379e-06 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatConvert             1 1.0 3.6716e-05 1.5 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatResidual          115 1.0 1.8372e-02 1.4 1.19e+07 1.1 9.2e+02 6.1e+02 0.0e+00  3  5  8  9  0   3  5  8  9  0  2504
      MatAssemblyBegin      30 1.0 6.0785e-03 5.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0     0
      MatAssemblyEnd        30 1.0 8.9223e-03 1.1 0.00e+00 0.0 1.5e+02 9.3e+01 8.8e+01  2  0  1  0  6   2  0  1  0  6     0
      MatGetRowIJ            1 1.0 8.3447e-06 1.8 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatCreateSubMats       3 1.0 2.5177e-04 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 6.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatGetOrdering         1 1.0 3.4571e-05 1.3 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatZeroEntries        18 1.0 4.1199e-04 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatFDColorCreate       6 1.0 4.7135e-04 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 1.2e+01  0  0  0  0  1   0  0  0  0  1     0
      MatFDColorSetUp        6 1.0 2.7599e-02 1.1 0.00e+00 0.0 0.0e+00 0.0e+00 4.8e+02  6  0  0  0 33   6  0  0  0 33     0
      MatFDColorApply       18 1.0 3.8840e-02 1.0 1.89e+07 1.1 3.0e+03 5.2e+02 2.8e+01  8  8 26 24  2   8  8 26 24  2  1887
      MatFDColorFunc       378 1.0 2.6123e-02 1.1 1.73e+07 1.1 3.0e+03 5.2e+02 0.0e+00  5  8 26 24  0   5  8 26 24  0  2566
      MatRedundantMat        3 1.0 3.0828e-04 1.2 0.00e+00 0.0 0.0e+00 0.0e+00 6.0e+00  0  0  0  0  0   0  0  0  0  0     0
      MatMPIConcateSeq       3 1.0 4.4107e-05 1.4 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
      DMCoarsen              5 1.0 1.9548e-03 1.0 0.00e+00 0.0 8.0e+01 2.2e+01 5.5e+01  0  0  1  0  4   0  0  1  0  4     0
      DMCreateInterp         5 1.0 6.3477e-03 1.0 5.76e+04 1.0 1.2e+02 6.6e+01 1.2e+02  1  0  1  0  8   1  0  1  0  8    36
      KSPSetUp              29 1.0 3.4931e-03 1.3 0.00e+00 0.0 0.0e+00 0.0e+00 7.6e+01  1  0  0  0  5   1  0  0  0  5     0
      KSPSolve               3 1.0 3.7720e-01 1.0 2.10e+08 1.1 1.1e+04 5.0e+02 1.3e+03 79 93 95 86 91  79 93 95 86 91  2166
      KSPGMRESOrthog       170 1.0 3.3011e-02 2.1 1.16e+07 1.1 0.0e+00 0.0e+00 1.7e+02  5  5  0  0 12   5  5  0  0 12  1363
      PCSetUp                3 1.0 5.2358e-02 1.0 5.09e+06 1.1 2.9e+03 3.2e+02 8.2e+02 11  2 25 14 56  11  2 25 14 56   367
      PCApply               23 1.0 3.0923e-01 1.0 1.94e+08 1.1 8.1e+03 5.4e+02 4.4e+02 63 86 69 69 30  63 86 69 69 30  2434
      ------------------------------------------------------------------------------------------------------------------------

      Memory usage is given in bytes:

      Object Type          Creations   Destructions     Memory  Descendants' Mem.
      Reports information only for process 0.

      --- Event Stage 0: Main Stage

                      SNES     1              1         1372     0.
                    DMSNES     6              6         4432     0.
            SNESLineSearch     1              1         1000     0.
                    Vector   231            231      4858432     0.
                    Matrix    56             56      4056840     0.
        Matrix FD Coloring     6              6      5749152     0.
          Distributed Mesh     6              6        31664     0.
                 Index Set   185            185       220700     0.
         IS L to G Mapping     6              6        17912     0.
         Star Forest Graph    12             12         9728     0.
           Discrete System     6              6         5572     0.
               Vec Scatter    29             29        97136     0.
             Krylov Solver    13             13       178448     0.
           DMKSP interface     6              6         3936     0.
            Preconditioner     8              8         7904     0.
                    Viewer     1              0            0     0.
      ========================================================================================================================
      Average time to get PetscTime(): 4.76837e-08
      Average time for MPI_Barrier(): 2.79903e-05
      Average time for zero size MPI_Send(): 1.04904e-05
      #PETSc Option Table entries:
      -da_refine 5
      -log_view
      -pc_type mg
      #End of PETSc Option Table entries
      Compiled without FORTRAN kernels
      Compiled with full precision matrices (default)
      sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
      Configure options: PETSC_DIR=/home/patrick/petsc-mb PETSC_ARCH=arch-ubuntu-mb-double-extra-opt --with-debugging=0 --with-precision=double --with-scalar-type=real --with-cc=gcc --with-cxx=g++ --with-fc=gfortran --download-mpich --COPTFLAGS="-g -O3 -march=native " --CXXOPTFLAGS="-g -O3 -march=native " --FOPTFLAGS="-g -O3 -march=native " --download-c2html --download-suitesparse --download-yaml --download-hdf5 --download-scalapack --download-metis --download-parmetis --download-mumps --download-superlu_dist --download-triangle --download-ctetgen --download-sundials --download-ml --download-exodusii --download-hdf5 --download-netcdf --download-pnetcdf
      -----------------------------------------
      Libraries compiled on 2018-07-27 13:01:14 on pdsbox
      Machine characteristics: Linux-4.13.0-39-generic-x86_64-with-Ubuntu-16.04-xenial
      Using PETSc directory: /home/patrick/petsc-mb
      Using PETSc arch: arch-ubuntu-mb-double-extra-opt
      -----------------------------------------

      Using C compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc    -Wall -Wwrite-strings -Wno-strict-aliasing -Wno-unknown-pragmas -fstack-protector -fvisibility=hidden
      Using Fortran compiler: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90   -Wall -ffree-line-length-0 -Wno-unused-dummy-argument
      -----------------------------------------

      Using include paths: -I/home/patrick/petsc-mb/include -I/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/include
      -----------------------------------------

      Using C linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpicc
      Using Fortran linker: /home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/bin/mpif90
      Using libraries: -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -lpetsc -Wl,-rpath,/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -L/home/patrick/petsc-mb/arch-ubuntu-mb-double-extra-opt/lib -Wl,-rpath,/usr/lib/gcc/x86_64-linux-gnu/5 -L/usr/lib/gcc/x86_64-linux-gnu/5 -Wl,-rpath,/usr/lib/x86_64-linux-gnu -L/usr/lib/x86_64-linux-gnu -Wl,-rpath,/lib/x86_64-linux-gnu -L/lib/x86_64-linux-gnu -lcmumps -ldmumps -lsmumps -lzmumps -lmumps_common -lpord -lpthread -lscalapack -lumfpack -lklu -lcholmod -lbtf -lccolamd -lcolamd -lcamd -lamd -lsuitesparseconfig -lsuperlu_dist -lml -lsundials_cvode -lsundials_nvecserial -lsundials_nvecparallel -llapack -lblas -lexodus -lnetcdf -lpnetcdf -lhdf5hl_fortran -lhdf5_fortran -lhdf5_hl -lhdf5 -lparmetis -lmetis -ltriangle -lm -lctetgen -lpthread -lyaml -lstdc++ -ldl -lmpifort -lmpi -lgfortran -lm -lgfortran -lm -lgcc_s -lquadmath -lstdc++ -ldl
      -----------------------------------------

  Compare the runtime for SNESSolve to the 1 processor case with multigrid. What is the speedup? Why is the speedup for multigrid lower than the speedup for the default solver?

## Nonlinear time dependent PDE on unstructured grid

WHAT THIS EXAMPLE DEMONSTRATES:

- Changing the default ODE integrator

- Handling unstructured grids

- Registering your own interchangeable physics and algorithm modules

FURTHER DETAILS:

- Mathematical description of the problem

- main program source code

- source code of physics modules

DO THE FOLLOWING:

- Compile `src/ts/tutorials/ex11.c`

      $ cd petsc/src/ts/tutorials
      $ make ex11

- Run simple advection through a tiny hybrid mesh

      $ mpiexec -n 1 ./ex11 -f ${PETSC_DIR}/share/petsc/datafiles/meshes/sevenside.exo

  Expected output:

        0  time        0  |x|    7.155
        1  time   0.7211  |x|    4.386
        2  time    1.442  |x|    1.968
        3  time    2.163  |x|    1.925
      CONVERGED_TIME at time 2.16333 after 3 steps

- Run simple advection through a small mesh with a Rosenbrock-W solver

      $ mpiexec -n 1 ./ex11 -f ${PETSC_DIR}/share/petsc/datafiles/meshes/sevenside.exo -ts_type rosw

  Expected output:

        0  time        0  |x|    7.155
        1  time  0.03608  |x|    7.045
        2  time   0.1088  |x|    6.807
        3  time   0.1805  |x|    6.558
        4  time   0.2501  |x|    6.303
        5  time   0.3189  |x|    6.044
        6  time   0.3954  |x|    5.748
        7  time   0.4793  |x|    5.419
        8  time   0.5692  |x|    5.064
        9  time   0.6635  |x|    4.694
       10  time   0.7596  |x|    4.322
       11  time   0.8563  |x|    3.955
       12  time   0.9553  |x|     3.59
       13  time     1.06  |x|    3.216
       14  time    1.176  |x|    2.823
       15  time    1.301  |x|    2.417
       16  time    1.439  |x|    2.002
       17  time    1.588  |x|    1.866
       18  time    1.748  |x|    1.902
       19  time    1.915  |x|     1.93
       20  time    2.082  |x|     1.95
      CONVERGED_TIME at time 2.08166 after 20 steps

- Run simple advection through a larger quadrilateral mesh of an annulus with least squares reconstruction and no limiting, monitoring the error

      $ mpiexec -n 4 ./ex11 -f ${PETSC_DIR}/share/petsc/datafiles/meshes/annulus-20.exo -monitor Error -advect_sol_type bump -petscfv_type leastsquares -petsclimiter_type sin

  Expected output:

        0  time        0  |x|   0.8708         Error [         0,         0] int          0
        1  time  0.07526  |x|   0.7882         Error [         0,0.09434159] int 0.05288794
        2  time   0.1505  |x|   0.7343         Error [         0, 0.2131413] int 0.09120994
        3  time   0.2258  |x|   0.6957         Error [         0, 0.2529671] int  0.1178105
        4  time    0.301  |x|   0.6728         Error [         0, 0.2209969] int  0.1510195
        5  time   0.3763  |x|   0.6484         Error [         0, 0.3418217] int  0.1742279
        6  time   0.4516  |x|   0.6259         Error [         0, 0.2836391] int  0.2025824
        7  time   0.5268  |x|   0.6001         Error [         0, 0.2995109] int  0.2355702
        8  time   0.6021  |x|   0.5671         Error [4.018435e-45, 0.4150825] int  0.2669753
        9  time   0.6774  |x|   0.5406         Error [1.972482e-39, 0.3607147] int  0.2858846
       10  time   0.7526  |x|   0.5171         Error [6.083192e-36,  0.448034] int  0.3098002
       11  time   0.8279  |x|    0.506         Error [3.572914e-33, 0.4455137] int  0.3356554
       12  time   0.9031  |x|   0.4947         Error [3.404038e-31, 0.4161921] int  0.3553281
       13  time   0.9784  |x|   0.4851         Error [2.737647e-29, 0.3831727] int  0.3701208
       14  time    1.054  |x|   0.4746         Error [1.595143e-27, 0.4870963] int   0.386669
       15  time    1.129  |x|   0.4618         Error [4.855084e-26, 0.4884613] int  0.4033437
       16  time    1.204  |x|   0.4501         Error [8.826408e-25, 0.4241629] int  0.4207896
       17  time    1.279  |x|   0.4399         Error [1.013076e-23, 0.4846543] int  0.4351298
       18  time    1.355  |x|   0.4309         Error [8.357122e-23, 0.5166439] int  0.4563886
       19  time     1.43  |x|   0.4192         Error [5.938047e-22, 0.4947022] int  0.4741966
       20  time    1.505  |x|    0.412         Error [3.987736e-21, 0.4909573] int  0.4931484
       21  time    1.581  |x|   0.4017         Error [2.328432e-20, 0.5286506] int  0.5038741
       22  time    1.656  |x|   0.3916         Error [1.188744e-19,  0.518947] int  0.5228681
       23  time    1.731  |x|   0.3824         Error [5.375705e-19, 0.5598767] int  0.5402542
       24  time    1.806  |x|   0.3774         Error [2.397704e-18, 0.5997036] int  0.5577958
       25  time    1.882  |x|   0.3712         Error [1.010082e-17, 0.5943566] int  0.5687406
       26  time    1.957  |x|   0.3641         Error [3.837194e-17, 0.5663827] int  0.5766598
       27  time    2.032  |x|   0.3569         Error [1.311105e-16, 0.5534062] int  0.5847664
      CONVERGED_TIME at time 2.03208 after 27 steps

  Compare turning to the error after turning off reconstruction.

- Run shallow water on the larger mesh with least squares reconstruction and minmod limiting, monitoring water Height (integral is conserved) and Energy (not conserved)

      $ mpiexec -n 4 ./ex11 -f ${PETSC_DIR}/share/petsc/datafiles/meshes/annulus-20.exo -physics sw -monitor Height,Energy -petscfv_type leastsquares -petsclimiter_type minmod

  Expected output:

        0  time        0  |x|    2.972        Height [         1,   2.97161] int   28.12971        Energy [       0.5,  4.415232] int   17.12186
        1  time   0.1597  |x|    2.125        Height [         1,  2.125435] int   28.12971        Energy [       0.5,  2.334901] int    17.2733
        2  time   0.3193  |x|    1.944        Height [         1,  1.943793] int   28.12971        Energy [       0.5,  1.904131] int   17.29508
        3  time    0.479  |x|    1.866        Height [         1,  1.865932] int   28.12971        Energy [       0.5,  1.741459] int   17.11151
        4  time   0.6386  |x|    1.677        Height [         1,  1.676699] int   28.12971        Energy [       0.5,  1.420746] int   16.96103
        5  time   0.7983  |x|    1.579        Height [         1,  1.578751] int   28.12971        Energy [       0.5,  1.266909] int    16.8442
        6  time   0.9579  |x|    1.534        Height [ 0.9066044,  1.533904] int   28.12971        Energy [ 0.4148315,  1.211645] int   16.73017
        7  time    1.118  |x|    1.522        Height [ 0.8458232,    1.5217] int   28.12971        Energy [ 0.3590109,  1.171816] int   16.64452
        8  time    1.277  |x|    1.495        Height [ 0.8767922,  1.494585] int   28.12971        Energy [ 0.3846352,  1.132207] int   16.59414
        9  time    1.437  |x|    1.462        Height [ 0.9860658,   1.46217] int   28.12971        Energy [  0.486527,  1.084667] int   16.56137
       10  time    1.597  |x|    1.424        Height [  1.000033,  1.424019] int   28.12971        Energy [ 0.5000325,  1.024201] int   16.51911
       11  time    1.756  |x|    1.381        Height [  1.000129,  1.381206] int   28.12971        Energy [ 0.5001287, 0.9675924] int   16.46646
       12  time    1.916  |x|    1.336        Height [  1.000441,  1.336399] int   28.12971        Energy [ 0.5004409, 0.9293328] int   16.41464
       13  time    2.076  |x|    1.308        Height [ 0.9827839,   1.30794] int   28.12971        Energy [ 0.4835028, 0.8976772] int   16.36243
      CONVERGED_TIME at time 2.07552 after 13 steps