Finite Element Approximation of Dirichlet Boundary Control for Elliptic PDEs on Two- and Three-Dimensional Curved Domains

Abstract

We consider the variational discretization of elliptic Dirichlet optimal control problems with constraints on the control. The underlying state equation, which is considered on smooth two- and three-dimensional domains, is discretized by linear finite elements taking into account domain approximation. The control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions, provided that the underlying mesh satisfies some additional condition. We confirm our analytical findings by numerical experiments.