Multigrid Methods for Discrete Fractional Sobolev Spaces
This work provides efficient preconditioners for fractional Sobolev spaces, which are important for solving coupled multiphysics problems, but the contribution is incremental as it adapts existing multigrid techniques.
The authors develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality, achieving a uniform bound on the condition number, and extend it to negative fractionality via left-right multiplication. Numerical experiments verify the theoretical results.
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.