petsc-3.12.4 2020-02-04
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# PCFieldSplitSetSchurFactType

sets which blocks of the approximate block factorization to retain in the preconditioner

### Synopsis

    #include "petscpc.h"
    PetscErrorCode  PCFieldSplitSetSchurFactType(PC pc,PCFieldSplitSchurFactType ftype)

Collective on PC

### Input Parameters

| pc | - the preconditioner context
| ftype | - which blocks of factorization to retain, PC_FIELDSPLIT_SCHUR_FACT_FULL is default

### Options Database

-pc_fieldsplit_schur_fact_type <diag,lower,upper,full> default is full -

### Notes

The FULL factorization is

      (A   B)  = (1       0) (A   0) (1  Ainv*B)  = L D U

      (C   E)    (C*Ainv  1) (0   S) (0     1  )

where S = E - C*Ainv*B. In practice, the full factorization is applied via block triangular solves with the grouping L*(D*U). UPPER uses D*U, LOWER uses L*D, and DIAG is the diagonal part with the sign of S flipped (because this makes the preconditioner positive definite for many formulations, thus allowing the use of KSPMINRES). Sign flipping of S can be turned off with PCFieldSplitSetSchurScale().

       If A and S are solved exactly

         *) FULL factorization is a direct solver.

         *) The preconditioned operator with LOWER or UPPER has all eigenvalues equal to 1 and minimal polynomial of degree 2, so KSPGMRES converges in 2 iterations.

         *) With DIAG, the preconditioned operator has three distinct nonzero eigenvalues and minimal polynomial of degree at most 4, so KSPGMRES converges in at most 4 iterations.

If the iteration count is very low, consider using KSPFGMRES or KSPGCR which can use one less preconditioner application in this case. Note that the preconditioned operator may be highly non-normal, so such fast convergence may not be observed in practice.

For symmetric problems in which A is positive definite and S is negative definite, DIAG can be used with KSPMINRES.

Note that a flexible method like KSPFGMRES or KSPGCR must be used if the fieldsplit preconditioner is nonlinear (e.g. a few iterations of a Krylov method is used to solve with A or S).

### References

| 1. | - Murphy, Golub, and Wathen, A note on preconditioning indefinite linear systems, SIAM J. Sci. Comput., 21 (2000).
| 2. | - Ipsen, A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23 (2001).

### See Also

PCFieldSplitGetSubKSP(), PCFIELDSPLIT, PCFieldSplitSetFields(), PCFieldSplitSchurPreType, PCFieldSplitSetSchurScale()

### Level

intermediate

### Location

src/ksp/pc/impls/fieldsplit/fieldsplit.c

### Implementations

PCFieldSplitSetSchurFactType_FieldSplit in src/ksp/pc/impls/fieldsplit/fieldsplit.c
Index of all PC routines
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Index of all manual pages