An Eulerian Finite Element Method for Elliptic Equations on Moving Surfaces

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, for example, two-phase incompressible flow problems. We give an analysis that shows that the method has optimal order of convergence both in the H1 and in the L2-norm. Results of numerical experiments are included that confirm this optimality.