From the summary: We present a mortar finite element formulation based on dual basis functions and on a special multigrid method. The starting point for our multigrid method is a symmetric positive definite system on the unconstrained product space. In addition, we introduce a new algorithm for the numerical solution of a nonlinear contact problem between two linear elastic bodies. It is shown that our method can be interpreted as an inexact Dirichlet-Neumann algorithm for nonlinear problems. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from a mortar discretization on non-matching triangulations with dual basis Lagrange multipliers. Numerical results illustrate the performance of our approach in two and three dimensions.