Efficient contact solvers based on domain decomposition techniques

This paper deals with the construction of efficient algorithms for the solution of the finite dimensional constrained minimization problem arising from the finite element discretization of contact problems. Dualization techniques have been used to decrease the problem size from the large number of unknowns in the domain to the much smaller number of inequalities at the boundary. The disadvantage of this direct Schur-complement approach is the need of the inversion of the stiffness matrix. Domain decomposition techniques meet very similar requirements. A global boundary value problem is decoupled into local subproblems, and one interface problem at the (coupling) boundary. Two complementary approaches are the Dirichlet method and the Neumann method. The first one requires preconditioners for local Dirichlet problems and for the interface problem in H+ , and extension operators from the boundary into the domain. The second one needs preconditioners for local Neumann problems, and for the interface problem in H− . Efficient multilevel algorithms for all components are available in literature. In this paper, it is shown how to use exactly these components for the construction of solvers for contact problems. New results for the analysis of convergence are presented. At least at uniformly refined meshes, we can prove optimal time complexity. Numerical results show high efficiency also on adaptively refined 3D meshes.