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.