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.

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.Key words. Elliptic boundary value problem, edge and vertex singularities, finite element method, anisotropic mesh grading, optimal control problem, discrete compactness property.

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.

Second-order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second-order PDEs in nondivergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. KEYWORDS finite element error estimates, PDEs in nondivergence form This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

This paper deals with error estimates for the finite element approximation of Neumann boundary control problems in polyhedral domains. Special emphasis is put on singularities contained in the solution as the computational domain has edges and corners. Thus, we use tailored regularity results in weighted Sobolev spaces which allow to derive sharp convergence results for locally refined meshes. The first main result is an optimal error estimate for linear finite element approximations on the boundary in the L 2 ()-norm for both quasi-uniform and isotropically refined meshes. Later, the approximations of Neumann control problems using the postprocessing approach are investigated, that is, first a fully discrete solution with piecewise linear state and co-state, and piecewise constant controls, is computed and afterwards, an improved control by a pointwise evaluation of the discrete optimality condition is obtained. It is shown that quadratic convergence up to logarithmic factors is achieved for this control approximation if either the singularities are weak enough or the sequence of meshes is refined appropriately.