Implementing Galerkin finite element methods for semilinear elliptic differential inclusions

This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of H₀ ¹. The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.