The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation

TL;DR

This paper investigates the spectral fractional Laplacian with measure-valued right-hand sides, establishing well-posedness, analyzing an optimal control problem, and developing finite element approximation schemes with proven convergence and error bounds.

Contribution

It introduces a weak formulation for the spectral fractional Laplacian with singular data, analyzes an optimal control problem, and proposes practical finite element schemes with rigorous error analysis.

Findings

Well-posedness of the weak formulation in fractional Sobolev spaces

Convergence of finite element scheme in energy norm

Error bounds for finite element approximations in L2 norm

Abstract

We consider the spectral definition of the fractional Laplace operator and study a basic linear problem involving this operator and singular forcing. In two dimensions, we introduce an appropriate weak formulation in fractional Sobolev spaces and prove that it is well-posed. As an application of these results, we analyze a pointwise tracking optimal control problem for fractional diffusion. We also develop a finite element scheme for the linear problem using continuous, piecewise linear functions, prove a convergence result in energy norm, and derive an error bound in $L^2(\Omega)$. Finally, we propose a practical scheme based on a diagonalization technique and derive an error bound in $L^2(\Omega)$ using a regularization argument.