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

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 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.

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

“…The application of a trace theorem to estimate the normal derivatives of the adjoint state in (4.28) yields suboptimal results only. In the case of the Poisson equation, and using standard finite element methods, optimal error estimates for the so called variational normal derivative can be found in [51,75,83]. However, proving sharp convergence rates for normal derivatives requires regularity results in weighted W k,∞ (Ω) spaces and non-standard duality arguments.…”

“…adjoint state and not-standard duality arguments to obtain estimates for the variational normal derivative; see also [51,75,83]. We are not aware of any similar study for HDG discretization.…”

Optimal Dirichlet control of partial differential equations on networks

“…Furthermore, sharp discretization error estimates for the approximate controls and the states in various norms are proved. Similar studies for related problems in bounded domains can be found in [1,8,40,46,52,68], where, e.g., linear convergence of the approximate controls in the L 2 -norm can be proved, provided that the exact solutions are sufficiently regular. A surprising observation in our setting is that the approximate controls u h converge even quadratically with respect to the mesh parameter h > 0, i.e., the estimate |u − u h | ≤ Ch 2 is satisfied.…”

“…Increasing the polynomial degree for the approximations will not lead to an improvement of the convergence rate. c) If a stronger norm for the regularization were used, e.g., H s (I; R n D ) with s > 0, then the optimal solution would be more regular leading to better approximation properties; see [46,68].…”

$L^\infty$ norm error estimates for HDG methods applied to the Poisson equation with an application to the Dirichlet boundary control problem

“…Another example appears in the problem of Dirichlet boundary control (DBC) of PDEs with L 2 (∂Ω)-regularization, where the normal derivative naturally arises in the discrete optimality system. Hence, the estimation of the error in the normal derivative plays an essential role in the error analysis of the DBC of PDEs, see [1,20,25,26,34] for more details. In recent papers where HDG methods have been sucessful applied to the DBC of PDEs ( [4,16,17,21,22]), the analysis for the control is optimal in the sense of regularity and suboptimal for other variables.…”