# Optimal operator preconditioning for pseudodifferential boundary problems

## TL;DR

This paper introduces an operator preconditioner for elliptic pseudodifferential equations that ensures mesh-independent condition numbers, improving the efficiency of solving discretized boundary problems across various geometries.

## Contribution

It extends Boggio's fractional Laplacian formula to develop a preconditioner applicable to general elliptic pseudodifferential equations on diverse domains, unifying and generalizing recent results.

## Key findings

- Condition numbers are mesh-independent with the new preconditioner.
- Numerical examples confirm the theoretical performance improvements.
- Preconditioner works effectively on various mesh types, including adaptive meshes.

## 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, N\'{e}d\'{e}lec and Urz\'{u}a-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.