A combined finite element method for elliptic problems posted in domains with rough boundaries

A combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries. Solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes, while good quality meshes often over-refine unnecessarily the interior of the domain. The basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size H≫h for other portions of the domain to reduce some unnecessary computational effort. The transmission conditions across the fine-coarse mesh interface are treated by the penalty technique. The key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form, which avoids the affection of the ratio H∕h in the error estimate. We prove a quasi-optimal convergence in terms of elements since there is no whole H2 regularity in the domain with rough boundaries. Numerical results are provided for elliptic equations in domains with non-oscillating or oscillating boundaries to illustrate the theoretical results.