Summary This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.

SUMMARYA framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.

In this article, a fully aggregation-based algebraic multigrid strategy is developed for nonlinear contact problems of saddle point type using a mortar finite element approach. While the idea of extending multigrid methods to saddle point systems can already be found, for example, in the context of Stokes and Oseen equations in literature, the main contributions of this work are (i) the development and open-source implementation of an interface aggregation strategy specifically suited for generating Lagrange multiplier aggregates that are required for coupling structural equilibrium equations with contact constraints and (ii) a review of saddle point smoothers in the context of constrained interface problems. The new interface aggregation strategy perfectly fits into an aggregation-based multigrid framework and can easily be combined with segregated transfer operators, which allow to preserve the saddle point structure on the coarse levels. Further analysis provides insight into saddle point smoothers applied to contact problems, while numerical experiments illustrate the robustness of the new method.We have implemented the proposed algorithm within the MueLu package of the open-source Trilinos project. Numerical examples demonstrate the robustness of the proposed method in complex dynamic contact problems as well as its scalability up to 23.9 million unknowns on 480 MPI ranks.

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This is the official user guide for the MUELU multigrid library in Trilinos version 11.12. This guide provides an overview of MUELU, its capabilities, and instructions for new users who want to start using MUELU with a minimum of effort. Detailed information is given on how to drive MUELU through its XML interface. Links to more advanced use cases are given. This guide gives information on how to achieve good parallel performance, as well as how to introduce new algorithms. Finally, readers will find a comprehensive listing of available MUELU options. Any options not documented in this manual should be considered strictly experimental.