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 $\mathbb{R}^n$ 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 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.