Multi-grid Methods for PDEs on Complicated Domains

Stefan Sauter
Universität Leipzig, Germany


In our talk we present composite finite elements for the solution of PDEs on complicated domains. Efficient solvers as multi-grid methods are based on a multi-scale discretization of the problem. However, for complicated domains and/or for problems with highly oscillating coefficients, coarse-scale discretizations are not available by using standard finite elements. The minimal dimension of classical finite element spaces is proportional to the number and size of geometric details. Composite finite elements overcome this difficulty by relaxing the condition ``a finite element mesh has to resolve the boundary'' by the condition ``the domain has to be covered by the finite element mesh''. Hence, the minimal dimension of composite finite element spaces is independent of the number and size of geometric details. The geometry is incorporated in the finite element functions in an appropriate way. The approximation property of composite finite elements can be proved in the same generality as for classical finite elements.

In our talk, we extend composite finite elements to problems with jumping coefficients and present an algebraic variant of the multi-grid algorithm based on composite finite elements: For a given fine-grid mesh and given fine-grid equation a hierarchy of coarse scale problems is set up along with appropriate prolongation operators in a black-box fashion so that the fine grid equations can be solved via a multi-grid algorithm. We illustrate the efficiency of the method by numerical experiments for the (indefinite) Helmholtz-problem.