A Multigrid Algorithm for the Mortar Finite Element Method

Abstract

The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar nite element discretization of second order elliptic boundary value problems. In order to establish the inf-sup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we revisit rst brie y the theoretical concept of the mortar nite element method. Employing suitable mesh-dependent norms we verify the validity of the LBB condition for the resulting mixed method and prove an L 2 error estimate. This is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid e ciency based on a smoother which is applied to the whole coupled system of equations. We conclude with several numerical tests of the proposed scheme which con rm the theoretical results and show the e ciency and the robustness of the method even in situations not covered by the theory.

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

“…In this section we shall apply the inequalities derived above in the context of the mortar finite element method ; see, e.g., [2,6,7,25]. The mortar method seems to be particularly well suited for problems with strong jumps in coefficients.…”

“…(see [6] for further background information). It is important to note that therefore each interface Γ kl appears only once 1 in the sum over S.…”

“…Here we follow the lines in [6,5,25] and adopt the above formulation as a mixed method. Thus the central issue is now to see under which circumstances…”

“…Of course, this depends on the norms for X h and M h which have yet to be specified. At first glance, · 1,Π seems to be a natural choice for X h while M h should be endowed with some sort of an H 1/2 norm on the skeleton S. However, this turns out to be inappropriate and we refer to [6] for the details.…”

“…The superscripts k, kl in h will be suppressed whenever the reference of h to an interface or subdomain is clear from the context. Guided by [6,5], we introduce the norms…”