# American Institute of Mathematical Sciences

March  2013, 18(2): 295-312. doi: 10.3934/dcdsb.2013.18.295

## Galerkin finite element methods for semilinear elliptic differential inclusions

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

Received  January 2012 Revised  May 2012 Published  November 2012

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.
Citation: Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
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