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A Semi-Analytic Diagonalization FEM for the Spectral Fractional Laplacian

Abner J. Salgado, Shane E. Sawyer

TL;DR

This work develops an exact diagonalization semi-analytic FEM for the spectral fractional Laplacian $(-\Delta)^s$ using the Caffarelli–Silvestre extension. By solving the extended-dimensional eigenproblem analytically, it decouples the problem into independent $\Omega$-problems and links the approach to Balakrishnan-type quadrature, enabling parallel computation without ill-conditioned eigenproblems. The authors prove an $L^2(\Omega)$ error decomposition into discretization, quadrature, and truncation components, and derive parameter choices that achieve $O(h^{2s})$ convergence. Numerical tests on an L-shaped and a circular domain confirm the predicted rates and demonstrate strong and weak MPI scalability. The method provides a stable, scalable pathway for fractional diffusion simulations with rigorous error control.

Abstract

We present a technique for approximating solutions to the spectral fractional Laplacian, which is based on the Caffarelli-Silvestre extension and diagonalization. Our scheme uses the analytic solution to the associated eigenvalue problem in the extended dimension. We show its relation to a quadrature scheme. Numerical examples demonstrate the performance of the method.

A Semi-Analytic Diagonalization FEM for the Spectral Fractional Laplacian

TL;DR

This work develops an exact diagonalization semi-analytic FEM for the spectral fractional Laplacian using the Caffarelli–Silvestre extension. By solving the extended-dimensional eigenproblem analytically, it decouples the problem into independent -problems and links the approach to Balakrishnan-type quadrature, enabling parallel computation without ill-conditioned eigenproblems. The authors prove an error decomposition into discretization, quadrature, and truncation components, and derive parameter choices that achieve convergence. Numerical tests on an L-shaped and a circular domain confirm the predicted rates and demonstrate strong and weak MPI scalability. The method provides a stable, scalable pathway for fractional diffusion simulations with rigorous error control.

Abstract

We present a technique for approximating solutions to the spectral fractional Laplacian, which is based on the Caffarelli-Silvestre extension and diagonalization. Our scheme uses the analytic solution to the associated eigenvalue problem in the extended dimension. We show its relation to a quadrature scheme. Numerical examples demonstrate the performance of the method.
Paper Structure (21 sections, 19 theorems, 130 equations, 3 figures, 3 tables)

This paper contains 21 sections, 19 theorems, 130 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Spectral Fractional Laplacian
  4. The Caffarelli-Silvestre Extension
  5. Truncation
  6. Tensorial Discretization
  7. Diagonalization
  8. Background
  9. A Semi-Analytic Approach
  10. Error Analysis
  11. The discrete Laplacian
  12. Error decomposition
  13. Discretization error
  14. Quadrature error: The Balakrishnan formula
  15. Error Analysis for $s=\tfrac{1}{2}$
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $f \in \mathbb{H}^{-s}(\Omega)$ and assume that $\mathcal{U} \in \mathring{H}_L^1(y^\alpha, \mathcal{C})$ solves eq:cs-weak-form. Then $u = \text{tr } \mathcal{U} \in \mathbb{H}^s(\Omega)$ solves eq:pde-problem in the sense that it satisfies eq:SolFracLapSeriesSense.

Figures (3)

  • Figure 1: Error in the $L^2(\Omega_L)$ norm versus the number of degrees of freedom using $\mathbb{Q}_1$ finite elements for $s=1/4$ and $s=3/4$ on uniformly refined meshes of $\Omega_L$.
  • Figure 2: Error in the $L^2(\Omega_C)$ norm versus the number of degrees of freedom using $\mathbb{Q}_1$ finite elements for $s=1/4$ and $s=3/4$ on uniformly refined meshes of $\Omega_C$.
  • Figure 3: Parallel performance for a simple test case.

Theorems & Definitions (39)

  • Theorem 2.1: extension
  • Proof 1
  • Proposition 2.2: truncation
  • Proof 2
  • Theorem 2.3: tensor products
  • Proof 3
  • Corollary 2.4: tensor products
  • Proof 4
  • Theorem 3.1: eigenpairs
  • Proof 5
  • ...and 29 more