The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917269777416192 |
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| author | Otarola, Enrique Salgado, Abner J. |
| author_facet | Otarola, Enrique Salgado, Abner J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation Otarola, Enrique Salgado, Abner J. Numerical Analysis Analysis of PDEs 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. |
| title | The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2602.11423 |