Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

Relaxed one-sided Lipschitz conditions play an important role when analyzing ordinary differential inclusions. They allow to derive a-priori estimates of solutions and convergence estimates for explicit and implicit time discretizations. In this paper we consider Galerkin finite element discretizations of semilinear elliptic inclusions that satisfy a relaxed one-sided Lipschitz condition. It is shown that solution sets of both, the continuous and the discrete system, are nonempty closed bounded and connected sets in H1-norm. Moreover, the solution sets of the Galerkin inclusion converge with respect to the Hausdorff distance measured in L-spaces. We also set up a full discretization of the Galerkin inclusion which uses a partitioning of the finite element space into cells and support functionals for measuring the residual of Galerkin approximations. An efficient implementation is developed that utilizes connectedness of the solution set and that is tested on a numerical example. ∗Supported by CRC 701 ’Spectral Structures and Topological Methods in Mathematics’, Bielefeld University.