S = A11 - A10 inv(A00) A01
PCLSC currently doesn't do anything with A11, so let's assume it is 0. The idea is that a good approximation to inv(S) is given by
inv(A10 A01) A10 A00 A01 inv(A10 A01)
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 A10 is a Laplacian, call it L. The current interface is to hang L and a preconditioning matrix Lp on the preconditioning 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) { assembly L } if (Lp) { assemble Lp }
With this, you should be able to choose LSC preconditioning, using e.g. ML's 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.
1. | - Elman, Howle, Shadid, Shuttleworth, and Tuminaro, Block preconditioners based on approximate commutators, 2006. | |
2. | - Silvester, Elman, Kay, Wathen, Efficient preconditioning of the linearized Navier Stokes equations for incompressible flow, 2001. |