A Finite Element Method for Elliptic Equations on Surfaces

Abstract

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. 1. Introduction. Moving hypersurfaces and interfaces appear in many physical processes, for example, in multiphase flows and flows with free surfaces. Certain mathematical models involve elliptic partial differential equations posed on such surfaces. This happens, for example, in multiphase fluids if one takes so-called surface active agents (surfactants) into account. These surfactants induce tangential surface tension forces and thus cause Marangoni phenomena [9,10]. Numerical simulations play an important role in a better understanding and prediction of processes involving this or other surface phenomena. In mathematical models surface equations are often coupled with other equations that are formulated in a (fixed) domain which contains the surface. In such a setting a common approach is to use a splitting scheme that allows to solve at each time step a sequence of simpler (decoupled) equations. In doing so one has to solve numerically at each time step an elliptic type of equation on a surface. The surface may vary from one time step to another and usually only some discrete approximation of the surface is available. A well-known finite element method for solving elliptic equations on surfaces, initiated by the paper [5], consists of approximating the surface by a piecewise polygonal surface and using a finite element space on a triangulation of this discrete surface, cf. [3,9]. If the surface is changing in time, then this approach leads to time-dependent triangulations and time-dependent finite element spaces. Implementing this requires substantial data handling and programming effort. Another approach has recently been introduced in [2]. The method in that paper applies to cases in which the surface is given implicitly by some level set function, and the key idea is to solve the partial differential equation on a narrow band around the surface. Unfitted finite element spaces on this narrow band are used for discretization.

A finite element method for surface PDEs: matrix properties

“…Descripton of the method. In this section we describe the finite element method from [7] for the three-dimensional case. The modifications needed for the two-dimensional case are obvious.…”

“…In [7] it is shown that this new method has optimal order of convergence in H 1 and L 2 norms. The analysis requires shape regularity of the outer triangulation, but does not require any type of shape regularity for discrete surface elements.…”

“…In section 2.1 we describe the finite element method that is introduced in [7]. In section 2.2 we give results of some numerical experiments.…”

“…If the surface is changing in time, then this approach leads to time-dependent triangulations and time-dependent finite element spaces. Implementing this requires substantial data handling and programming effort.In the recent paper [7] we introduced a new technique for the numerical solution of an elliptic equation posed on a hypersurface. The main idea is to use time-independent finite element spaces that are induced by triangulations of an "outer" domain to discretize the partial differential equation on the surface.…”

“…In the recent paper [7] we introduced a new technique for the numerical solution of an elliptic equation posed on a hypersurface. The main idea is to use time-independent finite element spaces that are induced by triangulations of an "outer" domain to discretize the partial differential equation on the surface.…”