A multigrid method based on the unconstrained product space for Mortar finite element discretizations

Abstract

The mortar finite element method allows the coupling of different discretizations across subregion boundaries. In the original mortar approach, the Lagrange multiplier space enforcing a weak continuity condition at the interfaces is defined as a modified finite element trace space. Here we present a new approach, where the Lagrange multiplier space is replaced by a dual space without losing the optimality of the a priori bounds. We introduce new dual spaces in 2D and 3D. Using the biorthogonality between the nodal basis functions of this Lagrange multiplier space and a finite element trace space, we derive an equivalent symmetric positive definite variational problem defined on the unconstrained product space. The introduction of this formulation is based on a local elimination process for the Lagrange multiplier. This equivalent approach is the starting point for the efficient iterative solution by a multigrid method. To obtain level independent convergence rates for the $\Cw$-cycle, we have to define suitable level dependent bilinear forms and transfer operators. Numerical results illustrate the performance of our multigrid method in 2D and 3D.