Local convergence of the FEM for the integral fractional Laplacian

TL;DR

This work establishes sharp local convergence rates for the finite element method applied to the integral fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$. By combining localization techniques, commutator estimates for the fractional operator, and a Caffarelli–Silvestre extension framework, the authors prove local $H^1$-norm and energy-norm error bounds that decompose into a local best-approximation error and a global weaker-norm slush term $\|u-u_h\|_{H^{s-1/2}(\Omega)}$, with corollaries giving optimal rates under mild local regularity on quasi-uniform meshes. The results underscore how local smoothness enhances local convergence while preserving global control via weaker norms, and they are complemented by numerical experiments confirming sharpness of the rates. The analysis relies on a shift-regularity assumption for a dual problem and on advanced tools such as commutator mappings and Beppo-Levi extension theory, offering practical guidance for local mesh refinement strategies near regions of interest.

Abstract

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.

Figures (5)

• Figure 1: Computational domain, local error computed on red subdomain.
• Figure 2: Local and global errors in the $L^2$- and $H^1$-norm for $s=0.1$.
• Figure 3: Local and global errors in the $L^2$- and $H^1$-norm, left: $s=0.3$; right: $s=0.5$.
• Figure 4: Local and global errors in the $L^2$- and $H^1$-norm, left: $s=0.7$; right: $s=0.9$.
• Figure 5: Local and global errors on graded meshes, left: $s=0.3$; right: $s=0.7$.

Theorems & Definitions (22)

• Remark 2.2
• Theorem 2.3
• Corollary 2.4
• Remark 2.5
• Remark 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3