Heterogeneous Multiscale FEM for Diffusion Problems on Rough Surfaces

Abstract

We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132]. For periodic surface roughness of scale $\varepsilon$ and amplitude $O(\varepsilon)$, the method converges at a robust rate, i.e., independent of $\varepsilon$, to the homogenized solution.