An Accurate Multigrid Solver for Computing Singular Solutions of Elliptic Problems

Harald Koestler

Lehrstuhl fuer Informatik 10 (Systemsimulation)
FAU Erlangen-Nuernberg
Cauerstrasse 6
D-91058 Erlangen

In this article we present a method to solve partial differential equations (PDE) containing generalized functions as source terms. Typical applications are point sources and sinks in porous media flow that are described by Dirac distributions or point loads and dipoles as source terms for electrostatic potentials. For analyzing the accuracy of such computations, standard techniques cannot be used, since they rely on global smoothness. This is both true for Sobolev space arguments for finite element based methods, and for continuity and differentiability arguments in finite difference analysis. At the singularity, the solution and its derivatives tend to infinity and therefore standard error norms will not even converge. We will demonstrate that these difficulties can be overcome by using other metrics to measure accuracy and convergence of the numerical solution. In the standard case our technique is equivalent to representing singular distributions by properly scaled finite element basis functions. Only minor modifications to the discretization and solver are necessary to obtain the same asymptotic accuracy and efficiency as for regular and smooth solutions.