Efficient numerical solution of Neumann problems on complicated domains

Abstract

We consider elliptic partial differential equations with Neumann boundary conditions on complicated domains. The discretization is performed by composite finite elements.
The a priori error analysis typically is based on precise knowledge of the regularity of the solution. However, the constants in the regularity estimates possibly depend critically on the geometric details of the domain and the analysis of their quantitative influence is rather involved.
Here, we consider a polyhedral Lipschitz domain Ω with a possibly huge number of geometric details ranging from size O(ε) to O(1). We assume that Ω is a perturbation of a simpler Lipschitz domain Ω. We prove error estimates where only the regularity of the partial differential equation on Ω is needed along with bounds on the norm of extension operators which are explicit in appropriate geometric parameters.
Since composite finite elements allow a multiscale discretization of problems on complicated domains, the linear system which arises can be solved by a simple multi-grid method. We show that this method converges at an optimal rate independent of the geometric structure of the problem.