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Functional a posteriori estimates for the fractional Laplacian problem

Alexander Nazarov, Sergey Repin

TL;DR

This work develops a fully computable a posteriori error framework for boundary value problems governed by the spectral fractional Laplacian. By leveraging the Stinga--Torrea extension to a local high-dimensional problem on Q, the authors derive an exact error identity and construct computable majorants and minorants that bound the energy-norm error of any energy-admissible approximation, independent of the specific numerical method used. The framework then translates extended-domain estimates to the original domain via trace relations, yielding practical, computable bounds for the original solution error. Numerical experiments with spectral-type approximations demonstrate the effectiveness of the estimates, particularly for s=1/2, and point to potential improvements with more sophisticated flux reconstructions. The approach provides robust error control tools applicable to a broad class of nonlocal problems that admit an extension to a local higher-dimensional problem.

Abstract

The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional operator. The derivation is based upon the Stinga--Torrea extension that transfers the corresponding nonlocal problem to a local problem of higher dimensionality. The estimates are fully computable and contain no conditions and constants depending on a method or mesh used to compute an approximation. They are valid for any energy admissible approximation of the extended problem.

Functional a posteriori estimates for the fractional Laplacian problem

TL;DR

This work develops a fully computable a posteriori error framework for boundary value problems governed by the spectral fractional Laplacian. By leveraging the Stinga--Torrea extension to a local high-dimensional problem on Q, the authors derive an exact error identity and construct computable majorants and minorants that bound the energy-norm error of any energy-admissible approximation, independent of the specific numerical method used. The framework then translates extended-domain estimates to the original domain via trace relations, yielding practical, computable bounds for the original solution error. Numerical experiments with spectral-type approximations demonstrate the effectiveness of the estimates, particularly for s=1/2, and point to potential improvements with more sophisticated flux reconstructions. The approach provides robust error control tools applicable to a broad class of nonlocal problems that admit an extension to a local higher-dimensional problem.

Abstract

The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional operator. The derivation is based upon the Stinga--Torrea extension that transfers the corresponding nonlocal problem to a local problem of higher dimensionality. The estimates are fully computable and contain no conditions and constants depending on a method or mesh used to compute an approximation. They are valid for any energy admissible approximation of the extended problem.
Paper Structure (7 sections, 6 theorems, 76 equations, 5 figures, 1 table)

This paper contains 7 sections, 6 theorems, 76 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. A posteriori estimates
  4. A posteriori error identity for the problem ${\mathcal{P}}^s_Q$
  5. Computable estimates of the norm $\mid\!\mid\!\mid \nabla_{xt}{\rm e}_w \mid\!\mid\!\mid$
  6. A posteriori error estimates for the problem ${\mathcal{P}}^s_\Omega$
  7. Example

Key Result

Lemma 1

For any $\widetilde{w} \in {\mathcal{V}}$ it holds

Figures (5)

  • Figure 1: Constants $C_s$ and $\kappa_s$
  • Figure 2: Functions $w$ and $w_t$ for $M=12$
  • Figure 3: Functions $w$ and $w_t$ for $M=16$
  • Figure 4: Disturbances of exact eigenvalues and eigenfunctions and the overall error for $M=N=12$.
  • Figure 5: Indexes of the Majorant

Theorems & Definitions (10)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Theorem 1
  • Corollary 2
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4