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Preconditioning in H$(\operatorname {div})$ and applications

Authors: Douglas N. Arnold, Richard S. Falk and R. Winther
Journal: Math. Comp. 66 (1997), 957-984
MSC (1991): Primary 65N55, 65N30
MathSciNet review: 1401938
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Abstract: We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I$- \operatorname {\mathbf {grad}}\operatorname {div}$. The natural setting for such problems is in the Hilbert space H$(\operatorname {div})$ and the variational formulation is based on the inner product in H$(\operatorname {div})$. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

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Additional Information

Douglas N. Arnold
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

R. Winther
Affiliation: Department of Informatics, University of Oslo, Oslo, Norway

Keywords: Preconditioner, mixed method, least squares, finite element, multigrid, domain decomposition
Received by editor(s): March 19, 1996
Received by editor(s) in revised form: April 19, 1996
Additional Notes: The first author was supported by NSF grants DMS-9205300 and DMS-9500672 and by the Institute for Mathematics and its Applications. The second author was supported by NSF grant DMS-9403552. The third author was supported by The Norwegian Research Council under grants 100331/431 and STP.29643.
Dedicated: Dedicated to Professor Ivo Babuška on the occasion of his seventieth birthday.
Article copyright: © Copyright 1997 American Mathematical Society