On the convergence in $H^1$-norm for the fractional Laplacian

TL;DR

This paper investigates the convergence of finite element solutions for the fractional Laplacian in the H^1-norm, providing error estimates on different mesh types for solutions with certain regularity.

Contribution

It establishes H^1-error estimates for finite element approximations of the fractional Laplacian, extending classical results to fractional-order problems.

Findings

Error estimates in H^1-norm derived for quasi-uniform meshes.

Error estimates in H^1-norm derived for graded meshes.

Analysis applies to solutions with regularity in H^1 due to problem conditions.

Abstract

We consider the numerical solution of the fractional Laplacian of index $s\in(1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\widetilde H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the ${\widetilde H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.