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An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

Jens Markus Melenk, Alexander Rieder

TL;DR

The paper studies numerical approximation of $\partial^{\gamma}_t u + \mathcal{L}^{\beta} u = f$ with Caputo time derivative and fractional spatial operator on analytic domains. It develops an exponentially convergent scheme by integrating a sinc-quadrature discretization of the Riesz-Dunford integral with $hp$ finite elements in space and $hp$ quadrature in time, and proves space-time and time-robust convergence even in the presence of startup singularities. A constructive hp-FEM framework is provided that resolves boundary-layer scales with exponential accuracy, including a cutoff-based initialization to satisfy right-space approximations. The authors validate the theory with 2D numerical experiments on the unit square, showing exponential convergence and computational efficiency through a minimized number of linear solves, extending hp-for-heat ideas to a functional-calculus-based fractional parabolic setting.

Abstract

We consider a space-time fractional parabolic problem. Combining a sinc-quadrature based method for discretizing the Riesz-Dunford integral with $hp$-FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an $hp$-quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times $t$, proving robustness with respect to startup singularities due to data incompatibilities.

An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

TL;DR

The paper studies numerical approximation of with Caputo time derivative and fractional spatial operator on analytic domains. It develops an exponentially convergent scheme by integrating a sinc-quadrature discretization of the Riesz-Dunford integral with finite elements in space and quadrature in time, and proves space-time and time-robust convergence even in the presence of startup singularities. A constructive hp-FEM framework is provided that resolves boundary-layer scales with exponential accuracy, including a cutoff-based initialization to satisfy right-space approximations. The authors validate the theory with 2D numerical experiments on the unit square, showing exponential convergence and computational efficiency through a minimized number of linear solves, extending hp-for-heat ideas to a functional-calculus-based fractional parabolic setting.

Abstract

We consider a space-time fractional parabolic problem. Combining a sinc-quadrature based method for discretizing the Riesz-Dunford integral with -FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an -quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times , proving robustness with respect to startup singularities due to data incompatibilities.
Paper Structure (12 sections, 18 theorems, 97 equations, 3 figures)

This paper contains 12 sections, 18 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model Problem and notation
  3. The Mittag-Leffler function
  4. Assumptions on the discretization in space
  5. The pure initial value problem
  6. The inhomogeneous problem
  7. $hp$-quadrature
  8. Analysis of the inhomogeneous problem
  9. The general result
  10. Space-time and time robust estimates
  11. $hp$-FEM
  12. Numerical Results

Key Result

Proposition 2.3

Let $z \in \mathscr{S}$ and $u_0 \in L^2(\Omega)$. Then, for $\alpha \in [0,1]$:

Figures (3)

  • Figure 2.1: Geometric configuration of Definition \ref{['def:domain_of_ellipticity']}
  • Figure 7.1: Example of a geometric mesh with 5 refinement layers.
  • Figure 7.2: Convergence at $t=1$ in the $L^2(\Omega)$-norm for compatible and incompatible data

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3: sinc_book
  • Proposition 3.4
  • ...and 37 more