Multigrid Methods for Discrete Fractional Sobolev Spaces

TL;DR

This paper develops an additive multigrid preconditioner for the fractional Laplacian applicable to both positive and negative fractional Sobolev spaces, providing uniform bounds and practical implementation insights.

Contribution

It introduces a novel multigrid preconditioner for the fractional Laplacian that works uniformly for positive and negative fractions, with detailed implementation guidance.

Findings

The preconditioner achieves a uniform bound on the condition number.

Numerical experiments verify the theoretical bounds.

The method effectively handles both positive and negative fractional Sobolev spaces.

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. In this work, we develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality, and show a uniform bound on the condition number. For the case of negative fractionality, we re-use the preconditioner developed for the positive fractionality and left-right multiply a regular Laplacian with a preconditioner with positive fractionality to obtain the desired negative fractionality. Implementational issues are outlined in details as the differences between the discrete operators and their corresponding matrices must be addressed when realizing these algorithms in code. We finish with some numerical experiments verifying the theoretical findings.