PCFieldSplitSetSchurFactType#

sets which blocks of the approximate block factorization to retain in the preconditioner [MGW00] and [Ips01]

Synopsis#

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

Collective

Input Parameters#

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

Options Database Key#

-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)
.vb
  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
.vb
  *) 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.

A flexible method like KSPFGMRES or KSPGCR, Flexible Krylov Methods, 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#

Ips01: Ilse CF Ipsen. A note on preconditioning nonsymmetric matrices. SIAM Journal on Scientific Computing, 23(3):1050–1051, 2001.
MGW00: Malcolm F Murphy, Gene H Golub, and Andrew J Wathen. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing, 21(6):1969–1972, 2000.