Finite element error estimates on the boundary with application to optimal control

This paper is concerned with the discretization of linear elliptic partial differential equations with Neumann boundary condition in polygonal domains. The focus is on the derivation of error estimates in the $L^2$-norm on the boundary for linear finite elements. Whereas common techniques yield only suboptimal results, a new approach in this context is presented which allows for quasi-optimal ones, i.e., for domains with interior angles smaller than $2\pi /3$ a convergence order two (up to a logarithmic factor) can be achieved using quasi-uniform meshes. In the presence of internal angles greater than $2\pi /3$ which reduce the convergence rates on quasi-uniform meshes, graded meshes are used to maintain the quasi-optimal error bounds.

This result is applied to linear-quadratic Neumann boundary control problems with pointwise inequality constraints on the control. The approximations of the control are piecewise constant. The state and the adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of the continuous optimal control are constructed which possess superconvergence properties. Based on the improved error estimates on the boundary and optimal regularity in weighted Sobolev spaces almost second order convergence is proven for the approximations of the continuous optimal control problem. Mesh grading techniques are again used for domains with interior angles greater than $2\pi /3$. A certain regularity of the active set is assumed.