Mass Lumping for the Optimal Control of Elliptic Partial Differential Equations

Abstract

The finite element discretization of a control constrained elliptic optimal control problem is studied. Control and state are discretized by higher order finite elements. The inequality constraints are only posed in the Lagrange points. The computational effort is significantly reduced by a new mass lumping strategy. The main contribution is the derivation of new a priori error estimates up to order $h^4$ on locally refined meshes. Moreover, we propose a new algorithmic strategy to obtain such highly accurate results. The theoretical findings are illustrated by numerical examples.