A Finite Element Method for Elliptic Equations on Surfaces

Abstract

In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the $H^1$- and in the $L^2$-norm. Results of numerical experiments are included that confirm this optimality.