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$hp$-FEM for the fractional heat equation

Jens Markus Melenk, Alexander Rieder

TL;DR

The paper tackles numerically solving the time-dependent fractional diffusion problem $\dot{u}+\mathcal{L}^s u=f$ on a Lipschitz domain by combining $hp$-FEM in space with $hp$-DG timestepping via the Caffarelli–Silvestre extension. A two-tier discretization is developed: a spatial semidiscretization using tensor-product $hp$-FEM in $\Omega\times(0,\infty)$ and a temporal discretization that achieves optimal or exponential rates under an abstract framework requiring stable liftings of the initial data and exponential-approximation of singular perturbations. For analytic 1D/2D geometries, the authors verify the assumptions on geometrically graded meshes and prove exponential convergence in space and favorable rates in time, with numerical experiments corroborating startup-singularity robustness and 2D polygonal cases. The results indicate that the proposed $hp$-FEM/$hp$-DG approach yields highly efficient and robust approximations for fractional diffusion problems in practical settings.

Abstract

We consider a time dependent problem generated by a nonlocal operator in space. Applying a discretization scheme based on $hp$-Finite Elements and a Caffarelli-Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-Discontinuous Galerkin based timestepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the original domain $Ω$.

$hp$-FEM for the fractional heat equation

TL;DR

The paper tackles numerically solving the time-dependent fractional diffusion problem on a Lipschitz domain by combining -FEM in space with -DG timestepping via the Caffarelli–Silvestre extension. A two-tier discretization is developed: a spatial semidiscretization using tensor-product -FEM in and a temporal discretization that achieves optimal or exponential rates under an abstract framework requiring stable liftings of the initial data and exponential-approximation of singular perturbations. For analytic 1D/2D geometries, the authors verify the assumptions on geometrically graded meshes and prove exponential convergence in space and favorable rates in time, with numerical experiments corroborating startup-singularity robustness and 2D polygonal cases. The results indicate that the proposed -FEM/-DG approach yields highly efficient and robust approximations for fractional diffusion problems in practical settings.

Abstract

We consider a time dependent problem generated by a nonlocal operator in space. Applying a discretization scheme based on -Finite Elements and a Caffarelli-Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using -Discontinuous Galerkin based timestepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the original domain .

Paper Structure

This paper contains 19 sections, 35 theorems, 171 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Discretization in space -- the semidiscrete scheme
  4. A related elliptic problem
  5. Discretization of the extended variable $y$
  6. $hp$-fem in 1d:
  7. The space $\mathbb{V}_h^{\mathcal{Y}}$:
  8. Discretization in $x$
  9. Returning to the semidiscretization
  10. Example of a space $\mathbb{V}_h^{\mathcal{X}}$: $hp$-FEM in 1D and 2D
  11. Discretization in $t$ -- the fully discrete scheme
  12. Practical aspects
  13. Numerical Results
  14. Smooth solution
  15. Singular solution
  16. ...and 4 more sections

Key Result

Theorem 3.2

The operator $-\mathcal{L}_h^s$ is the generator of an analytic semigroup on $\left(\mathbb{V}_h^{\mathcal{X}}, \left\|\cdot\right\|_{L^2(\Omega)}\right)$.

Figures (4)

  • Figure 3.1: Geometric configuration of Definition \ref{['def:domain_of_ellipticity']}
  • Figure 3.2: The geometric situation in the proof of Theorem \ref{['thm:hp_fem_resolves_scales']} (for $c=0$).
  • Figure 5.1: Convergence rate in the case of non-matching initial condition
  • Figure 5.2: Convergence for the $2d$ and smooth cases

Theorems & Definitions (85)

  • Remark 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.6
  • Remark 3.7
  • Definition 3.8
  • Lemma 3.10
  • ...and 75 more