Abstract: 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 Raviart-Thomas 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. The Dirichlet problem is then extended to a variational problem on the whole nonconforming ansatz space. It can be shown that this is equivalent to a standard mortar coupling between conforming and Crouzeix-Raviart finite elements where the Lagrange multiplier lives on the side of the Crouzeix-Raviart elements. We note that the 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.