#include "petscpc.h" PetscErrorCode PCFieldSplitSetSchurFactType(PC pc,PCFieldSplitSchurFactType ftype)Collective on PC
| pc | - the preconditioner context | |
| ftype | - which blocks of factorization to retain, PC_FIELDSPLIT_SCHUR_FACT_FULL is default |
(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).
| 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). |