Analysis of a sinc-Galerkin Method for the Fractional Laplacian

TL;DR

This paper analyzes a sinc-Galerkin method for solving the fractional Dirichlet problem, establishing convergence and optimal error estimates without requiring strong regularity assumptions on solutions.

Contribution

It reinterprets a previous sinc-based collocation method as a nonconforming Galerkin method, providing a rigorous convergence analysis and error bounds.

Findings

Proves convergence of the sinc-Galerkin method.

Establishes optimal order of convergence.

Removes regularity assumptions on solutions.

Abstract

We provide the convergence analysis for a sinc-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of an earlier article by the same authors, where the authors presented a sinc-function based method to solve fractional PDEs. While the original method was formulated as a collocation method, we show that the same method can be interpreted as a nonconforming Galerkin method, giving access to abstract error estimates. Optimal order of convergence is shown without any unrealistic regularity assumptions on the solution.