287 RWTH Publication No: 47088        2008        IGPM287.pdf
TITLE A Finite Element Method for Surface PDES: Matrix Properties
AUTHORS Maxim A. Olshanskii, Arnold Reusken
ABSTRACT We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. The main idea of this method 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. It has been proved that the method has optimal order of convergence both in the H1 and in the L2 -norm. In this paper we address linear algebra aspects of this new finite element method. In particular the conditioning of the mass and stiffness matrix is investigated. For the two-dimensional case we present an analysis which proves that the (effective) spectral condition number of both the diagonally scaled mass matrix and the diagonally scalled stiffness matrix behaves like h−2, where h is the mesh size of the outer triangulation.
KEYWORDS surface, interface, finite element, level set method, two-phase flow
DOI 10.1007/s00211-009-0260-4
PUBLICATION Numerische Mathematik
Volume 114, Issue 3 , pp 491-520 (2009)