Convergence rates of the fractional to the local Dirichlet problem

2024/08/06 by Bungert, Leon, del Teso, Félix · 1 citation
#35A15 #35B30 #35B40 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.03299

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 √(1-s), while for less regular data the rate is of order √((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∈(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.

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