A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems

Nonconforming domain decomposition methods and their application to the numerical simulation of non-linear multibody contact problems play an important role in many applications in mechanics. To handle the non-linearity of the contact conditions, we apply a primal-dual active set strategy based on dual Lagrange multipliers. Combining this method with an optimal multigrid for the resulting linear algebraic problems and using inexact strategies, our algorithm yields an efficient iterative solver. Furthermore, we establish, under some regularity assumptions on the solution, optimal convergence orders for the discretization errors for the displacement and the Lagrange multiplier for linear and quadratic finite element spaces; we combine quadratic finite elements with linear and quadratic dual Lagrange multipliers. Several numerical examples confirm our theoretical results. In the last section, we extend our algorithm to a dynamic non-linear multibody contact problem.