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# PCLSC

Preconditioning for Schur complements, based on Least Squares Commutators [EHS+06] [SEKW01]

## Options Database Key

- -pc_lsc_commute - Whether to commute the LSC preconditioner in the style of Olshanskii

- -pc_lsc_scale_diag - Whether to scale \(BB^T\) products. Will use the inverse of the diagonal of \(Qscale\) or \(A\) if the former is not provided

## Notes

This preconditioner will normally be used with `PCFIELDSPLIT` to precondition the Schur complement, but it can be used for any Schur complement system. Consider the Schur complement

\[ S = A11 - A10 A00^{-1} A01 \]

`PCLSC` currently doesn’t do anything with \(A11\), so let’s assume it is 0. The idea is that a good approximation to \(S^{-1}\) is given by

\[ (A10 A01)^{-1} A10 A00 A01 (A10 A01)^{-1} \]

The product \(A10 A01\) can be computed for you, but you can provide it (this is usually more efficient anyway). In the case of incompressible flow, \(A10 A01\) is a Laplacian; call it \(L\). The current interface is to compose \(L\) and a matrix from which to construct a preconditioner \(Lp\) on the matrix.

If you had called `KSPSetOperators`(ksp,S,Sp), \(S\) should have type `MATSCHURCOMPLEMENT` and \(Sp\) can be any type you like (`PCLSC` doesn’t use it directly) but should have matrices composed with it, under the names “LSC_L” and “LSC_Lp”. For example, you might have setup code like this

       PetscObjectCompose((PetscObject)Sp,"LSC_L",(PetscObject)L);
       PetscObjectCompose((PetscObject)Sp,"LSC_Lp",(PetscObject)Lp);

And then your Jacobian assembly would look like

       PetscObjectQuery((PetscObject)Sp,"LSC_L",(PetscObject*)&L);
       PetscObjectQuery((PetscObject)Sp,"LSC_Lp",(PetscObject*)&Lp);
       if (L) { assemble L }
       if (Lp) { assemble Lp }

With this, you should be able to choose LSC preconditioning, using e.g. the `PCML` algebraic multigrid to solve with L

       -fieldsplit_1_pc_type lsc -fieldsplit_1_lsc_pc_type ml

Since we do not use the values in Sp, you can still put an assembled matrix there to use normal preconditioners.

## References

[EHS+06]

H.C. Elman, V.E. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput., 27(5):1651–1668, 2006.

[SEKW01]

D. Silvester, H. Elman, D. Kay, and A. Wathen. Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow. Journal of Computational and Applied Mathematics, 128(1-2):261–279, 2001.

## See Also

KSP: Linear System Solvers, `PCCreate()`, `PCSetType()`, `PCType`, `PC`, `Block_Preconditioners`, `PCFIELDSPLIT`, `PCFieldSplitGetSubKSP()`, `PCFieldSplitSetFields()`, `PCFieldSplitSetType()`, `PCFieldSplitSetIS()`, `PCFieldSplitSetSchurPre()`, `MatCreateSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementSetSubMatrices()`, `MatSchurComplementUpdateSubMatrices()`, `MatSchurComplementSetAinvType()`, `MatGetSchurComplement()`

## Level

intermediate

## Location

src/ksp/pc/impls/lsc/lsc.c

Index of all PC routines
Table of Contents for all manual pages
Index of all manual pages