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SUMMARYWe consider the numerical simulation of non-linear multi-body contact problems in elasticity on complex three-dimensional geometries. In the case of warped contact boundaries and non-matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the nonpenetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non-conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element-based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non-linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach.

The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics. Problem setting 2.1 Parameterized eigenvalue problems in computational mechanicsLet the computational domain Ω ⊂ R d , with d = 2, 3, be bounded and polygonal. As an elliptic eigenvalue model problem, we consider the linear elasticity case. But all our results also hold true for more general elliptic systems. Then, the eigenvalue problem

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