Optimal operator preconditioning for pseudodifferential boundary problems

Heiko Gimperlein, Jakub Stocek, Carolina Urzúa-Torres

Research output: Contribution to journalArticlepeer-review

Abstract

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain Ω, where Ω is either in R n or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives a unified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, Nédélec and Urzúa-Torres. We also study the increasing relevance of the regularity assumptions on the mesh with the order of the operator. Numerical examples validate our theoretical findings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptively generated meshes.

Original languageEnglish
Pages (from-to)1-41
Number of pages41
JournalNumerische Mathematik
Volume148
Issue number1
Early online date15 Apr 2021
DOIs
Publication statusPublished - May 2021

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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