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Iterative solution of a coupled mixed and standard Galerkin discretization method for elliptic problems
Author(s) -
Lazarov Raytcho D.,
Pasciak Joseph E.,
Vassilevski Panayot S.
Publication year - 2000
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 - 10.1002/1099-1506(200101/02)8:1<13::aid-nla225>3.0.co;2-g
Subject(s) - preconditioner , mathematics , conjugate gradient method , domain decomposition methods , finite element method , discretization , iterative method , saddle point , linear system , positive definite matrix , mathematical analysis , eigenvalues and eigenvectors , mathematical optimization , geometry , physics , quantum mechanics , thermodynamics
In this paper, we consider approximation of a second‐order elliptic problem defined on a domain in two‐dimensional Euclidean space. Partitioning the domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed finite element approximation on one subdomain with a standard finite element approximation on the other. In this paper, we study the iterative solution of the resulting linear system of equations. This system is symmetric and indefinite (of saddle‐point type). The stability estimates for the discretization imply that the algebraic system can be preconditioned by a block diagonal operator involving a preconditioner for H (div) (on the mixed side) and one for the discrete Laplacian (on the finite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem defined only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gradient iteration. Copyright © 2001 John Wiley & Sons, Ltd.