Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization

TL;DR

This paper provides precise error estimates for finite element approximations of variational normal derivatives and Dirichlet boundary control problems, highlighting the influence of domain geometry on convergence rates.

Contribution

It introduces a novel analysis linking convergence rates to the geometry of polygonal domains using weighted Sobolev and Hölder spaces.

Findings

Convergence rates depend on the largest opening angle of polygonal domains.

Numerical experiments confirm the sharpness of the theoretical error estimates.

The analysis improves understanding of finite element approximation accuracy in irregular domains.

Abstract

This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy regularization. The regularity of the solution is carefully carved out exploiting weighted Sobolev and H\"older spaces. This allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains. Numerical experiments confirm that the derived convergence rates are sharp.