Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization
Provides rigorous error bounds for a class of PDE-constrained optimization problems on polygonal domains, which is incremental for the numerical analysis community.
This paper derives sharp error estimates for finite element approximations of variational normal derivatives and Dirichlet control problems with energy regularization, linking convergence rates to the largest opening angle in polygonal domains. Numerical experiments confirm the sharpness of the rates.
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 Hölder 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.