Constructive approximation on graded meshes for the integral fractional Laplacian

TL;DR

This paper develops graded mesh techniques for approximating solutions to the integral fractional Laplacian, achieving optimal convergence rates by leveraging Sobolev regularity estimates in Lipschitz domains.

Contribution

It introduces a greedy algorithm for constructing graded meshes that attain quasi-optimal convergence rates for fractional Laplacian problems.

Findings

Optimal Sobolev regularity estimates for solutions in Lipschitz domains.

Construction of graded bisection meshes using a greedy algorithm.

Derivation of quasi-optimal convergence rates for piecewise linear approximations.

Abstract

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-\Delta)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.