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

Enrique Otarola, Abner J. Salgado

TL;DR

This work addresses the spectral fractional Laplacian $(-\Delta)^s$ with measure-valued right-hand sides in 2D, formulating a robust weak problem in fractional Sobolev spaces and proving well-posedness via the BNB theorem. It couples this analysis to a pointwise tracking optimal control problem and develops both an ideal and a practical finite element discretization; the ideal scheme attains optimal $L^2$-rates under restrictive regularity, while the practical diagonalization-based scheme, augmented by regularization, achieves the same $L^2$-convergence rate with implementable computations. Key contributions include a rigorous convergence theory for the discrete problems, $L^2$-error estimates, and a scalable numerical strategy based on a Balakrishnan-type representation and eigen-decomposition. The results are complemented by numerical illustrations that confirm the anticipated behavior of fractional solutions and demonstrate the viability of the proposed methods for nonlocal PDEs with singular forcing. Overall, the paper advances analysis, discretization, and computation for spectral fractional diffusion with measure data, with direct relevance to optimal control and nonlocal modeling.

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.

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

TL;DR

This work addresses the spectral fractional Laplacian with measure-valued right-hand sides in 2D, formulating a robust weak problem in fractional Sobolev spaces and proving well-posedness via the BNB theorem. It couples this analysis to a pointwise tracking optimal control problem and develops both an ideal and a practical finite element discretization; the ideal scheme attains optimal -rates under restrictive regularity, while the practical diagonalization-based scheme, augmented by regularization, achieves the same -convergence rate with implementable computations. Key contributions include a rigorous convergence theory for the discrete problems, -error estimates, and a scalable numerical strategy based on a Balakrishnan-type representation and eigen-decomposition. The results are complemented by numerical illustrations that confirm the anticipated behavior of fractional solutions and demonstrate the viability of the proposed methods for nonlocal PDEs with singular forcing. Overall, the paper advances analysis, discretization, and computation for spectral fractional diffusion with measure data, with direct relevance to optimal control and nonlocal modeling.

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 . Finally, we propose a practical scheme based on a diagonalization technique and derive an error bound in using a regularization argument.
Paper Structure (15 sections, 11 theorems, 129 equations, 2 figures)

This paper contains 15 sections, 11 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary remarks
  3. The spectral fractional Laplacian
  4. Analysis of the problem
  5. The pointwise tracking optimal control problem
  6. Discretization
  7. The discrete Laplacian
  8. An ideal discrete problem
  9. Convergence
  10. Rates of convergence
  11. A practical scheme
  12. The diagonalization scheme
  13. A practical scheme
  14. Suitable regularizations
  15. Numerical Illustrations

Key Result

Proposition 2.1

We have that with equivalent norms. Moreover, if $r \in (1,2)$, then ${\mathbb H}^r(\Omega) = H_0^1(\Omega) \cap H^r(\Omega)$, with equivalent norms.

Figures (2)

  • Figure 1: The left panel shows the solution to the fractional Laplacian ($s=0.65$) with the right hand side given by \ref{['eq:Curve']}. For comparison, the right panel shows the solution to the Laplacian with the same right hand side.
  • Figure 2: The left panel shows the solution to the fractional Laplacian ($s=0.65$) with the right hand side given by \ref{['eq:Dirac']}. For comparison, the right panel shows the solution to the Laplacian whit the same right hand side.

Theorems & Definitions (25)

  • Proposition 2.1: characterization of $\mathbb{H}^r(\Omega)$ for $r \in (0,2)$
  • Remark 3.1: on the role of $s$ and $\theta$ in \ref{['eq:WeakFormulation']}
  • Theorem 3.2: BNB
  • Proof 1
  • Remark 3.3: scaling
  • Theorem 4.1: existence and uniqueness
  • Proof 2
  • Theorem 4.2: optimality conditions
  • Proof 3
  • Remark 5.1: coefficient vectors
  • ...and 15 more