Abstract:
We consider some applications of domain decomposition methods with nonmatching meshes. Of special interest are coupling techniques for overlapping and nonoverlapping decompositions based on the weak mortar surface coupling. We focus on the coupling of mechanical structures, and we consider different material laws (linear and nonlinear) and material parameters. To illustrate the locking effect in case of nearly incompressible materials for standard conforming discretization schemes, we compare numerical simulations. A rigorous analysis can be given for generalized Hu-Washizu three-field formulations. This abstract setting includes many choices of finite elements which are widely used in the engineering community but also provides new possibilities. We address numerical stability problems in case of curvilinear interfaces and propose suitable modifications. These modifications can be locally carried out and are of higher order such that the asymptotic rate of convergence is preserved. However, for coarse meshes a considerably large difference can be observed for the original and modified approach. In addition to the discretization, we discuss briefly relevant issue of multigrid method applied to the solution of coupled problems. The key ingredients are nonnested spaces and free Dirichlet boundaries. Special transfer operators based on multiplicative structures will be given providing optimal multigrid convergence. All theoretical results are illustrated by numerical simulations in two and three dimensions including multigrid convergences rates, semismooth Newton iterations for variational inequalities and a priori and a posteriori convergence rates.
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