Solving Frictional Contact Problems with Multigrid Efficiency

The construction of fast and reliable solvers for contact problems with friction is even nowadays a challenging task. It is well known that contact problems with Coulomb friction have the weak form of a quasi-variational inequality [KO88, HHNL88, NJH80]. For small coefficients of friction, a solution can be obtained by means of a fixed point iteration in the boundary stresses [NJH80]. This fixed point approach is often used for the construction of numerical methods, since in each iteration step only a constrained convex minimization problem has to be solved [DHK02, LPR91]. Unfortunately, the convergence speed of the discrete fixed point iteration deteriorates for smaller meshsizes. Here, we present a nonlinear multigrid method which removes the outer fixed point iteration and gives rise to a highly efficient solution method for frictional contact problems with Coulomb friction and other local friction laws in 2 and 3 space dimensions. The numerical cost is comparable to frictionless contact problems. Our method is based on monotone multigrid methods, see [KK01], and does not require any regularization of the non penetration condition or of the friction law. Therefore, the results are highly accurate. Using the basis transformation given in [WK00], our method can also be applied to two body contact problems.