An accurate multigrid solver for computing singular solutions of elliptic problems

Köstler H, Rüde U (2006)

Publication Type: Journal article

Publication year: 2006


Publisher: Wiley-Blackwell

Book Volume: 13

Pages Range: 231-249

Journal Issue: 2-3


DOI: 10.1002/nla.478


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 δ-functions or point loads and dipoles as source terms for electrostatic potentials. For analysing the accuracy of such computations, standard techniques cannot be used, since they rely on global smoothness. This is true for both 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 a mesh-dependent representation of the singular sources constructed by appropriate smoothing and 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. Copyright © 2006 John Wiley & Sons, Ltd.

Authors with CRIS profile

Related research project(s)

How to cite


Köstler, H., & Rüde, U. (2006). An accurate multigrid solver for computing singular solutions of elliptic problems. Numerical Linear Algebra With Applications, 13(2-3), 231-249.


Köstler, Harald, and Ulrich Rüde. "An accurate multigrid solver for computing singular solutions of elliptic problems." Numerical Linear Algebra With Applications 13.2-3 (2006): 231-249.

BibTeX: Download