petsc-master 2020-10-22
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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 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).

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

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