The Coupling of Mixed and Conforming Finite Element Discretizations

In this paper, we introduce and analyze a special mortar finite element method. We restrict ourselves to the case of two disjoint subdomains, and use RaviartThomas finite elements in one subdomain and conforming finite elements in the other. In particular, this might be interesting for the coupling of different models and materials. Because of the different role of Dirichlet and Neumann boundary conditions a variational formulation without a Lagrange multiplier can be presented. It can be shown that no matching conditions for the discrete finite element spaces are necessary at the interface. Using static condensation, a coupling of conforming finite elements and enriched nonconforming Crouzeix-Raviart elements satisfying Dirichlet boundary conditions at the interface is obtained. Then the Dirichlet problem is extended to a variational problem on the whole nonconforming ansatz space. In this step a piecewise constant Lagrange multiplier comes into play. By eliminating the local cubic bubble functions, it can be shown that this is equivalent to a standard mortar coupling between conforming and Crouzeix-Raviart finite elements. Here the Lagrange multiplier lives on the side of the Crouzeix-Raviart elements. And in contrast to the standard mortar P1/P1 coupling the discrete ansatz space for the Lagrange multiplier consists of piecewise constant functions instead of continuous piecewise linear functions. We note that the piecewise constant Lagrange multiplier represents an approximation of the Neumann boundary condition at the interface. Finally, we present some numerical results and sketch the ideas of the algorithm. The arising saddle point problems is be solved by multigrid techniques with transforming smoothers. The mortar methods have been introduced recently and a lot of work in this field has been done during the last few years; cf., e.g., [1, 4, 5, 14, 15]. For the construction of efficient iterative solvers we refer to [2, 3, 20, 21]. The concepts of a posteriori error estimators and adaptive refinement techniques have also been generalized to mortar methods on nonmatching grids; see e.g. [13, 22, 25, 24].