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Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab

TL;DR

This paper proves that the integral fractional Laplacian on a 1D bounded interval admits weighted analytic regularity for analytic data, and uses this to establish exponential convergence of hp-FEM on geometric boundary-refined meshes. The analysis relies on the Caffarelli-Silvestre extension to a weighted local PDE, global and interior regularity via difference quotients and Caccioppoli-type estimates, and weighted embeddings to derive analytic bounds. Consequently, the hp-FEM on a geometrically refined mesh achieves exponential convergence in the energy norm, with rates independent of the polynomial degree when the mesh layering is chosen proportional to the degree. Numerical experiments on (-1,1) corroborate the theory, showing exponential energy-norm convergence and rates consistent with the predicted geometric progression in the mesh layers.

Abstract

We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity result, we prove exponential convergence of the hp-FEM on geometric boundary-refined meshes.

Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

TL;DR

This paper proves that the integral fractional Laplacian on a 1D bounded interval admits weighted analytic regularity for analytic data, and uses this to establish exponential convergence of hp-FEM on geometric boundary-refined meshes. The analysis relies on the Caffarelli-Silvestre extension to a weighted local PDE, global and interior regularity via difference quotients and Caccioppoli-type estimates, and weighted embeddings to derive analytic bounds. Consequently, the hp-FEM on a geometrically refined mesh achieves exponential convergence in the energy norm, with rates independent of the polynomial degree when the mesh layering is chosen proportional to the degree. Numerical experiments on (-1,1) corroborate the theory, showing exponential energy-norm convergence and rates consistent with the predicted geometric progression in the mesh layers.

Abstract

We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity result, we prove exponential convergence of the hp-FEM on geometric boundary-refined meshes.
Paper Structure (11 sections, 8 theorems, 51 equations, 1 figure)

This paper contains 11 sections, 8 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model problem and main results
  3. The fractional Laplacian
  4. Weighted analytic regularity
  5. Exponential convergence of $hp$-FEM
  6. Regularity results
  7. The Caffarelli-Silvestre extension
  8. Global regularity
  9. Interior regularity
  10. Exponential convergence of $hp$-FEM
  11. Numerical example

Key Result

theorem 1

Let the data $f\in C^\infty(\overline{\Omega})$ satisfy, for constants $C_f$, $\gamma_f >0$, Let $u$ solve eq:weakform. Then, there is $\gamma$ (depending only on $\gamma_f$, $s$, $\Omega$) such that for any $\varepsilon>0$ there exists a constant $C_{\varepsilon}>0$ (depending on $C_f$, $s$, $\Omega$, $\varepsilon$) such that In terms of the space ${\mathcal{B}}^1_\beta := \{u \in L^2(\Omega)\,

Figures (1)

  • Figure 1: Exponential energy norm error convergence of $hp$-FEM on geometric mesh with grading factor $\sigma = 0.6$ for $s \in \{0.3, 0.5, 0.7\}$, $\Omega = (-1,1)$, $f = 1$. Left: $hp$-FEM based on $S^{p,1}_0({\mathcal{T}}^{L}_{geo,\sigma})$ with $p = L$. Right: $hp$-FEM based on subspace $\widetilde{S}^L \subset S^{L,1}_0({\mathcal{T}}^{L}_{geo,\sigma})$.

Theorems & Definitions (17)

  • theorem 1
  • definition 1
  • theorem 2
  • lemma 1
  • proof : Sketch
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • ...and 7 more