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 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 H
1 and in the L
2-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.
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 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 H
1 and in the L
2-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.
- Content Type Journal Article
- DOI 10.1007/s00211-009-0260-4
- Authors
- Maxim A. Olshanskii, Moscow State M.V. Lomonosov University Department of Mechanics and Mathematics 119899 Moscow Russia
- Arnold Reusken, RWTH-Aachen University Institut für Geometrie und Praktische Mathematik 52056 Aachen Germany
- Journal Numerische Mathematik
- Online ISSN 0945-3245
- Print ISSN 0029-599X
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