An Auxiliary Space Preconditioner for Fractional Laplacian of Negative Order

TL;DR

This paper introduces an auxiliary space preconditioner for the fractional Laplacian of negative order, complementing existing methods for positive order, and demonstrates its spectral equivalence and effectiveness through numerical experiments.

Contribution

It develops a new auxiliary space preconditioner for the fractional Laplacian with negative order, expanding the toolkit for solving coupled multiphysics problems involving fractional Sobolev spaces.

Findings

Preconditioners are spectrally equivalent to $(- riangle)^{-s}$ for $s extless 0$.

The proposed preconditioners require fractional $H( ext{div})$ operators with positive fractionality.

Numerical experiments verify the theoretical spectral equivalence and effectiveness.

Abstract

Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian, $(-\Delta)^s$, with $s \in [-1,1]$. Previously, additive multigrid preconditioners for the case when $s \geq 0$ have been proposed. In this work we complement this construction with auxiliary space preconditioners suitable when $s \leq 0$. These preconditioners are shown to be spectrally equivalent to $(-\Delta)^{-s}$, but requires preconditioners for fractional $H(\operatorname{div})$ operators with positive fractionality. We design such operators based on an additive multigrid approach. We finish with some numerical experiments, verifying the theoretical results.