Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains

Abstract

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.

Error estimates for the finite element approximation of bilinear boundary control problems

“…For this approach we have to assume more regularity for the exact solution and thus, we restrict our considerations to two-dimensional domains with sufficiently smooth boundary. Under this assumption we show the optimal convergence rate of 2 − ε with arbitrary ε > 0 which is the rate one would also expect in the case of linear quadratic boundary control problems and smooth solutions [2,3,29] (even with h −ε replaced by |ln h|, where h is the maximal element diameter of the finite element mesh). The proof relies on the non-expansivity of the projection onto the feasible set as well as sharp error estimates for the state and adjoint state in L 2 (Γ).…”

Finite Volume Element Method for Solving the Elliptic Neumann Boundary Control Problems

“…However, there are only a few published results on the finite volume element method for the distributed optimal control problems. In [15] [16], the authors discussed distributed optimal control problems governed by elliptic equations by using the finite volume element methods. The variational discretization approach is used to deal with the control and the error estimates are obtained in some norms.…”

Weak discrete maximum principle of finite element methods in convex polyhedra

“…By utilizing the weak maximum principle, Schatz also established the stability of the Ritz projection in L ∞ and W 1,∞ norms. Such stability results have a wide range of applications, for example to pointwise error estimates of finite element methods for parabolic problems [16,20,21], Stokes systems [3], nonlinear problems [10,11,22], obstacle problems [6], optimal control problems [1,2], to name a few. As far as we know, [25] is the only paper that establishes weak maximum principle and L ∞ stability estimate (without the logarithmic factor) for the Ritz projection on nonsmooth domains.…”