A Multigrid Algorithm for the Mortar Finite Element Method

Abstract

The objective of this paper is to develop and analyze a multigrid algorithm for the system of equations arising from the mortar finite 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 first revisit briefly the theoretical concept of the mortar finite element method. Employing suitable mesh-dependent norms we verify the validity of the Ladyzhenskaya--Babuska--Brezzi (LBB) condition for the resulting mixed method and prove an L2 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 efficiency 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 confirm the theoretical results and show the efficiency and the robustness of the method even in situations not covered by the theory.