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In this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth complementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples.

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SUMMARYA common approach for the numerical simulation of non-linear multi-body contact problems is the use of Lagrange multipliers to model the contact conditions. The stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes. By this, the spurious algorithmic oscillations in the multiplier do not occur any more, which facilitates the application of the primal-dual active set strategy to dynamical contact problems. The new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost. In addition the conservation properties of the underlying algorithm are transferred to the modified mass version. Different numerical examples for frictional two-body contact problems illustrate the improvement in the results for the contact stresses.

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Abstract. Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. In this paper, we analyze the Nitsche method for the Lamé operator, establish a priori error estimates and compare this method with the mortar method using dual Lagrange multiplier spaces. Both methods can be applied to non matching triangulations. We use a multigrid algorithm to solve the algebraic systems. Although we have a mesh dependent bilinear form, optimal W-cycle convergence rates can be obtained. Numerical results for the two methods with linear and quadratic finite elements illustrate the performance and flexibility of these non conforming discretization techniques.Key words. mortar finite elements, Nitsche methods, domain decomposition, non matching triangulations, multigrid methods, mesh dependent bilinear form, linear elasticity AMS subject classifications. 65N15, 65N30, 65N551. Introduction. For the efficient numerical approximation of boundary value problems, domain decomposition methods are often used and provide powerful techniques. In particular, non conforming techniques provide a more flexible approach. Here, we consider two non overlapping domain decomposition techniques and non matching triangulations at the inner boundaries of the subdomains. Non matching triangulations can arise, e.g., if the meshes on the different subdomains are generated independently from each other, or if one subdomain is sliding and no remeshing strategy is used. To use highly different mesh sizes on the different subdomains is of special interest in the case of corner singularities or in the case of different material parameters. Then in general, no strong continuity condition at the inner boundaries can be imposed on the discrete solution. There are several different approaches to deal with this situation. Here, we focus on mortar [Ben99, BMP93, BMP94] and Nitsche [Nit71, BH99, HP00, BHS01, HN01] techniques. In the case of mortar methods, the strong pointwise continuity condition is replaced by a weaker integral condition. There are several different but equivalent variational formulations. One possibility is to introduce the flux on the skeleton as additional unknown and to write the discrete variational formulation as a saddle point problem. To obtain a stable and optimal scheme, the discrete Lagrange multiplier space has to satisfy a uniform inf-sup condition and suitable approximation properties. Elimination of the Lagrange multiplier yields a positive definite formulation on the constrained mortar space. In the case of dual Lagrange multiplier spaces [Woh01], the elimination can be carried out locally. On the other hand, Nitsche techniques yield a symmetric and positive definite bilinear form on the unconstrained product space depending of the actual mesh size. To obtain a stable scheme, the jump of the solution has to be penalized. The mesh dependent bilinear form depends on a fixed parameter which has to be chosen large enough. In both situations, the resulting linear system ...

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