Optimal operator preconditioning for pseudodifferential boundary problems

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain Omega, where Omega 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, Nedelec and Urzua-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.

Automated summary

In this paper, an operator preconditioner for general elliptic pseudodifferential equations is proposed, achieving condition numbers independent of mesh size and choice of bases in low-order Galerkin discretizations. The study provides a unified and independent proof for specific cases of operators, emphasizing the increasing relevance of mesh regularity assumptions with the order of the operator. Numerical examples validate theoretical findings and demonstrate the performance of the proposed preconditioner on various types of meshes.