Köstler H, Rüde U (2004)

**Publication Type:** Conference contribution

**Publication year:** 2004

**Publisher:** Springer-Verlag

**Edited Volumes:** Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

**Series:** Lecture Notes in Computer Science

**City/Town:** Berlin

**Book Volume:** 3039

**Pages Range:** 410-417

**Conference Proceedings Title:** Computational Science - ICCS 2004

**Event location:** Krakau

Generalized functions occur in many practical applications as source terms in partial differential equations. Typical examples are 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. At the singularity, the solution tends to infinity and therefore standard error norms will not even converge. In this article 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 point-wise 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. It is also possible to integrate these techniques into a multigrid solver exploiting additional techniques for improving the accuracy, such as Richardson and τ-Extrapolation. © Springer-Verlag Berlin Heidelberg 2004.

Harald Köstler
Lehrstuhl für Informatik 10 (Systemsimulation)
Ulrich Rüde
Lehrstuhl für Informatik 10 (Systemsimulation)

**APA:**

Köstler, H., & Rüde, U. (2004). Extrapolation Techniques for Computing Accurate Solutions of Elliptic Problems with Singular Solutions. In *Computational Science - ICCS 2004* (pp. 410-417). Krakau: Berlin: Springer-Verlag.

**MLA:**

Köstler, Harald, and Ulrich Rüde. "Extrapolation Techniques for Computing Accurate Solutions of Elliptic Problems with Singular Solutions." *Proceedings of the International Conference on Computational Science, Krakau* Berlin: Springer-Verlag, 2004. 410-417.

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