Accurate multigrid techniques for computing singular solutions of elliptic problems.

Ulrich Ruede

Institute of Computer Science X,
Universitaet Erlangen-Nuernberg
Cauerstr. 6
D-91058 Erlangen, Germany

Harald Koestler, Marcus Mohr


Generalized functions occur in many practical applications as source terms in partial differential applications. Typical examples are point sources and sinks in porous media flow that are described by Dirac delta functions or point loads and dipoles as source terms inducing electrostatic potentials. We are particularly interested in bioelectric field computations where the source terms are modeled by dipoles and where the computational goal is to locate dipole sources as accurately as possible from electroencephalographic measurements.

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 tends to infinity and therefore standard error norms will not even converge.

In this presentation we will demonstrate that these difficulties can be overcome by using other metrics to measure accuracy and convergence of the numerical solution. Only minor modifications to the discretization and solver are necessary to obtain the same asymptotic accuracy and efficiency as for regular and smooth solutions. In particular, no adaptive refinement is necessary and it is also unnecessary to use techniques which make use of the analytic knowledge of the singularity. Our method relies simply on a mesh-size dependent representation of the singular sources constructed by appropriate smoothing. It can be proved that the pointwise accuracy is of the same order as in the regular case. The error coefficient depends on the location and will deteriorate when approaching the singularity where the error estimate breaks down. Our approach is therefore useful for accurately computing the global solution, except in a small neighborhood of the singular points. In the talk we will demonstrate how these techniques can be integrated into a multigrid solver exploiting additional techniques for improving the accuracy, such as Richardson and tau-Extrapolation. The talk is a follow-up to a paper presented in Copper Mountain in 1987.