A multigrid algorithm for the mortar finite element method

The objective of this paper is to develop and analyze a multigrid algorithm for the system of equations arising from the mortar finite element discret ization of second order elliptic boundary value problems. In order to estab lish 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 suit able mesh-dependent norms we verify the validity of the Ladyzhenskaya-Babus ka-Brezzi (LBB) condition for the resulting mixed method and prove an L-2 e rror estimate. This is the key for establishing a suitable approximation pr operty for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid efficiency based on a smoother whi ch is applied to the whole coupled system of equations. We conclude with se veral numerical tests of the proposed scheme which confirm the theoretical results and show the efficiency and the robustness of the method even in si tuations not covered by the theory.