Convergence rates of the fractional to the local Dirichlet problem

TL;DR

This paper establishes explicit convergence rates in fractional Sobolev norms for solutions of fractional Dirichlet problems approaching the classical local Dirichlet problem as the fractional parameter s approaches 1, using variational methods.

Contribution

It provides the first non-asymptotic convergence rates for fractional to local Dirichlet problems in Sobolev spaces, including cases with variable right-hand sides.

Findings

Rate of order √(1-s) for regular boundary data

Rate of order √((1-s)|log(1-s)|) for less regular data

Error estimates valid for all s in (0,1)

Abstract

We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb R^d)$-norm for the solution of the fractional Dirichlet problem to the solution of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary values we get a rate of order $\sqrt{1-s}$, while for less regular data the rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the right hand side depends on $s$, and our error estimates are true for all $s\in(0,1)$. The proofs use variational arguments to deduce rates in the fractional Sobolev norm from energy estimates between the fractional and the standard Dirichlet energy.