Thomas Apel

Thomas Apel, Serge Nicaise, Johannes Pfefferer

The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the $L^2(Ω)$-norm with order $1/2$ in convex domains but has a reduced convergence order in non-convex domains although the solution remains to be contained in $H^{1/2}(Ω)$. The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases also with non-convex domains. Numerical experiments confirm the theoretical results.

Thomas Apel, Mariano Mateos, Johannes Pfefferer et al.

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided.

Thomas Apel, Ariel L. Lombardi, Max Winkler

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H^1(Ω)- and L^2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

Thomas Apel, Johannes Pfefferer, Sergejs Rogovs et al.

This paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in case of smooth domains. As a remedy the use of local mesh refinement near the corners is investigated. In order to prove quasi-optimal a priori error estimates regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^2 \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.

Thomas Apel, Katharina Lorenz, Johannes Pfefferer

The Stokes problem with non-homogeneous Dirichlet boundary condition is solved numerically using conforming discretizations and an approximation of the boundary datum in the corresponding trace space. Optimal discretization error estimates are derived. The theory accounts for the influence of corner singularities in the case of a non-convex domain. Several variants of the boundary data approximation are discussed. Moreover, the case of boundary data with very low regularity is studied, where a weak solution does not exist. The well-posedness of the very weak solution is investigated, and optimal discretization error estimates are derived. Numerical tests confirm the theory. The compatibility condition for the boundary data is not necessary for well-posedness of the weak and very weak formulations but it ensures that the solution satisfies the continuity equation in the distributional sense. In the same spirit, the compatibility condition is not necessary for the approximating boundary data; a good approximation of the original boundary data is important.

Thomas Apel, Mariano Mateos, Johannes Pfefferer et al.

A linear quadratic Dirichlet control problem posed on a possibly non-convex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetskii) spaces. In particular, it is proved that in the presence of control constraints, the optimal control is continuous despite the non-convexity of the domain.

Thomas Apel, Serge Nicaise, Johannes Pfefferer

The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges with order $1/2$ in convex domains but has a reduced convergence order in non-convex domains. As a remedy, a dual variant of the singular complement method is proposed. The error order of the convex case is retained. Numerical experiments confirm the theoretical results.

Thomas Apel, Serge Nicaise, Johannes Pfefferer

Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in $L^2$ only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to $2π$.