Finite element error estimates for normal derivatives on boundary concentrated meshes

TL;DR

This paper analyzes finite element approximations of normal derivatives for elliptic PDEs, demonstrating improved convergence rates on boundary concentrated meshes and applying results to boundary control problems.

Contribution

It introduces boundary concentrated meshes for better normal derivative approximation and proves doubled convergence rates, enhancing numerical analysis of boundary control problems.

Findings

Standard meshes yield near first-order convergence in L2 norm.

Boundary concentrated meshes double the convergence order.

Application to Dirichlet boundary control problems demonstrates practical benefits.

Abstract

This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson equation with homogeneous Dirichlet boundary conditions and use standard linear finite elements for its discretization. The underlying domain is assumed to be polygonal but not necessarily convex. Approximations of the normal derivatives are introduced in a standard way as well as in a variational sense. On general quasi-uniform meshes, one can show that these approximate normal derivatives possess a convergence rate close to one in $L^2$ as long as the singularities due to the corners are mild enough. Using boundary concentrated meshes, we show that the order of convergence can even be doubled in terms of the mesh parameter while increasing the complexity of the discrete problems only by a small factor. As an application, we use these results for the numerical analysis of Dirichlet boundary control problems, where the control variable corresponds to the normal derivative of some adjoint variable.