MatCreateSchurComplement#

Creates a new Mat that behaves like the Schur complement of a matrix

Synopsis#

#include "petscksp.h" 
PetscErrorCode MatCreateSchurComplement(Mat A00, Mat Ap00, Mat A01, Mat A10, Mat A11, Mat *S)

Collective

Input Parameters#

A00 - the upper-left block of the original matrix A = [A00 A01; A10 A11]
Ap00 - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A00^{-1}
A01 - the upper-right block of the original matrix A = [A00 A01; A10 A11]
A10 - the lower-left block of the original matrix A = [A00 A01; A10 A11]
A11 - (optional) the lower-right block of the original matrix A = [A00 A01; A10 A11]

Output Parameter#

S - the matrix that behaves as the Schur complement S = A11 - A10 ksp(A00,Ap00) A01

Notes#

The Schur complement is NOT explicitly formed! Rather, this function returns a virtual Schur complement that can compute the matrix-vector product by using formula S = A11 - A10 A^{-1} A01 for Schur complement S and a KSP solver to approximate the action of A^{-1}.

All four matrices must have the same MPI communicator.

A00 and A11 must be square matrices.

MatGetSchurComplement() takes as arguments the index sets for the submatrices and returns both the virtual Schur complement (what this returns) plus a sparse approximation to the Schur complement (useful for building a preconditioner for the Schur complement) which can be obtained from this matrix with MatSchurComplementGetPmat()

Developer Note#

The API that includes MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementGetPmat() should be refactored to remove redundancy and be clearer and simpler.