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

\[\left(\begin{array}{cc} A & B \\ C & E \\ \end{array}\right) = \left(\begin{array}{cc} I & 0 \\ C A^{-1} & I \\ \end{array}\right) \left(\begin{array}{cc} A & 0 \\ 0 & S \\ \end{array}\right) \left(\begin{array}{cc} I & A^{-1}B \\ 0 & I \\ \end{array}\right) = L D U,\]

where \( S = E - C A^{-1} B \). In practice, the full factorization is applied via block triangular solves with the grouping \(L(DU)\). upper uses \(DU\), lower uses \(LD\), 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

1 - full factorization is a direct solver.
2 - 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.
3 - 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.