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

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.