Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains

摘要

This article deals with error estimates for the finite element approximation of Neumann boundary control problems in polyhedral domains. Special emphasis is put on singularities contained in the solution, as the computational domain has edges and corners. Thus, we use regularity results in weighted Sobolev spaces, which allow to derive sharp convergence results for locally refined meshes. The first main result is an optimal error estimate for linear finite element approximations on the boundary in the L-2(Gamma)-norm for both quasi-uniform and isotropically refined meshes. Later, the approximations of Neumann control problems using the postprocessing approach are investigated, that is, first a fully discrete solution with piecewise linear state and co-state, and piecewise constant controls, is computed and afterwards, an improved control by a pointwise evaluation of the discrete optimality condition is obtained. It is shown that quadratic convergence up to logarithmic factors is achieved for this control approximation if either the singularities are weak enough or the sequence of meshes is refined appropriately.