M. Jung and T. D. Todorov

On the convergence factor in multilevel methods for solving 3D elasticity problems

Preprint SFB393/04-13, Sonderforschungsbereich 393, TU Chemnitz

The constant $\gamma$ in the strengthened Cauchy-Bunyakowski-Schwarz inequality is a basic tool for the construction of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.

Key words: strengthened Cauchy-Schwarz-Buniakowski inequality, linear elasticity problem, finite element method, multigrid method

AMS subject classification: 65N30, 65N55, 65N25.