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arXiv · 2602.11423

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

Enrique Otarola·Abner J. Salgado
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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 $L^2(Ω)$. Finally, we propose a practical scheme based on a diagonalization technique and derive an error bound in $L^2(Ω)$ using a regularization argument.

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Enrique Otarola, Abner J. Salgado. 2026-02-11. The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation. https://arxiv.org/abs/2602.11423

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