Multiscale Problems; Meshes and Solvers

Abstract

The efficient solution of large scaled finite element (f. e.) equations from industrial problems is a challenging task. Typically, such problems are quite complex due to the multiscale behavior of the geometry and the data. This is the reason why standard algorithms for mesh generation and solution are inefficient or can even fail. In this context, robust methods for treating multiscale problems are needed. The mesh generator should produce rather fine meshes catching the geometry with an asymptotically optimal complexity. Such a generator is proposed in two and tree dimensions. Usually, the meshes for geometries with microscales contain a large number of nodes. Thus, fast solvers for the f. e. equations are needed, where no hierarchical mesh structure is available. Such solvers are presented for the Potential and the Linear Elasticity Problems in two and three space dimensions within the framework of Algebraic Multigrid Methods. Together, these algorithms form an efficient tool for the Numerical Simulation of Multiscale Problems and can be easily integrated in existing f. e. codes including commercial codes.