Premium
Projected Schur complement method for solving non‐symmetric systems arising from a smooth fictitious domain approach
Author(s) -
Haslinger J.,
Kozubek T.,
Kučera R.,
Peichl G.
Publication year - 2007
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.550
Subject(s) - schur complement , mathematics , biconjugate gradient stabilized method , krylov subspace , saddle point , domain decomposition methods , domain (mathematical analysis) , generalized minimal residual method , boundary value problem , finite element method , multigrid method , boundary (topology) , mathematical analysis , matrix (chemical analysis) , linear system , geometry , partial differential equation , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , thermodynamics
Abstract This paper deals with a fast method for solving large‐scale algebraic saddle‐point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non‐symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle‐point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non‐symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non‐projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright © 2007 John Wiley & Sons, Ltd.