Galerkin finite element methods for semilinear elliptic differential inclusions

Wolf-Jüergen Beyn; Janosch Rieger

doi:10.3934/dcdsb.2013.18.295

Article Contents

2013, Volume 18, Issue 2: 295-312. Doi: 10.3934/dcdsb.2013.18.295

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Galerkin finite element methods for semilinear elliptic differential inclusions

Wolf-Jüergen Beyn^1, and
Janosch Rieger^2,

1.
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld
2.
Institut für Mathematik, Universität Frankfurt, Postfach 111932, D-60054 Frankfurt a.M.

Received: December 31, 2011

Revised: April 30, 2012

Published: February 2013

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Abstract

In this paper we consider Galerkin finite element discretizations of semilinear elliptic differential inclusions that satisfy a relaxed one-sided Lipschitz condition. The properties of the set-valued Nemytskii operators are discussed, and it is shown that the solution sets of both, the continuous and the discrete system, are nonempty, closed, bounded, and connected sets in $H^1$-norm. Moreover, the solution sets of the Galerkin inclusion converge with respect to the Hausdorff distance measured in $L^p$-spaces.

Keywords:

Elliptic partial differential inclusions,
finite element methods,
uncertainty quantification,
set-valued numerical analysis.

Mathematics Subject Classification: Primary: 35R70, 65L60; Secondary: 35J57 34A60.

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