A finite element method for surface PDEs: matrix properties

We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. Themain 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 problem in an outer domain that contains the surface, for example, two-phase 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 the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h−3| ln h| and h−2| ln h|, respectively, where h is the mesh size of the outer triangulation. Mathematics Subject Classification (2000) 58J32 · 65N15 · 65N30 · 76D45 · 76T99 Partially supported by the the Russian Foundation for Basic Research through the projects 08-01-00159 and 09-01-00115. This work was supported by the German Research Foundation through SFB 540. M. A. Olshanskii Department of Mechanics and Mathematics, Moscow State M.V. Lomonosov University, 119899 Moscow, Russia e-mail: [email protected] A. Reusken (B) Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, 52056 Aachen, Germany e-mail: [email protected]