Optimal operator preconditioning for pseudodifferential boundary problems

Abstract

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\Omega$, where $\Omega$ is either in ${\mathbb{R}}^n$ 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.

30 pages, 19 figures, to appear in Numerische Mathematik

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• Spectral results for mixed problems and fractional elliptic operators
• Space-time adaptive finite elements for nonlocal parabolic variational inequalities
• Uniform preconditioners for problems of positive order
• Uniform preconditioners of linear complexity for problems of negative order
• On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Cited by in corpus (1)

• Fractional Dissipative PDEs