Robust BPX Preconditioner for Fractional Laplacians on Bounded Lipschitz Domains

TL;DR

This paper introduces a robust BPX preconditioner for the fractional Laplacian on Lipschitz domains, ensuring uniformly bounded condition numbers across different grid types and fractional powers, applicable to spectral and censored variants.

Contribution

The paper develops and analyzes a new BPX preconditioner that maintains stability and efficiency for fractional Laplacian problems on complex domains.

Findings

Condition numbers are uniformly bounded for various grid types and fractional powers.

The preconditioner is effective for spectral and censored fractional Laplacians.

The approach enhances computational stability for fractional PDEs.

Abstract

We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power. The results apply also to the spectral and censored fractional Laplacians.