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Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes

Natalia Kopteva, Sean Kelly

TL;DR

This work develops sharp pointwise-in-time error bounds for L1-type discretizations of time-fractional (Caputo) semilinear and quasilinear parabolic equations on graded temporal meshes, addressing initial-singularity behavior with exponent $σ$. A novel stability framework for the discrete fractional derivative on quasi-graded meshes is established and extended to arbitrary grading $r≥1$, enabling precise truncation-control for nonlinear discretizations and enabling rigorous error estimates for both ODE-like reductions and full PDE discretizations. The analysis covers semidiscretizations in time and full discretizations in space via finite differences and finite elements, and it extends to quasilinear subdiffusion with non-self-adjoint operators, supported by numerical experiments that confirm optimal rates and the impact of mesh grading. Overall, the paper fills a gap by providing sigma-dependent, pointwise temporal error bounds on graded meshes for fractional subdiffusion, with practical implications for reliable and efficient numerical simulations.

Abstract

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^σ$ for any fixed $σ\in (0,1) \cup (1,2)$. The L1 scheme in time is combined with a general class of discretizations for the semilinear term. For such discretizations, we obtain sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading. Both semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretcal findings are illustrated by numerical experiments.

Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes

TL;DR

This work develops sharp pointwise-in-time error bounds for L1-type discretizations of time-fractional (Caputo) semilinear and quasilinear parabolic equations on graded temporal meshes, addressing initial-singularity behavior with exponent . A novel stability framework for the discrete fractional derivative on quasi-graded meshes is established and extended to arbitrary grading , enabling precise truncation-control for nonlinear discretizations and enabling rigorous error estimates for both ODE-like reductions and full PDE discretizations. The analysis covers semidiscretizations in time and full discretizations in space via finite differences and finite elements, and it extends to quasilinear subdiffusion with non-self-adjoint operators, supported by numerical experiments that confirm optimal rates and the impact of mesh grading. Overall, the paper fills a gap by providing sigma-dependent, pointwise temporal error bounds on graded meshes for fractional subdiffusion, with practical implications for reliable and efficient numerical simulations.

Abstract

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order are considered, solutions of which exhibit a singular behaviour at an initial time of type for any fixed . The L1 scheme in time is combined with a general class of discretizations for the semilinear term. For such discretizations, we obtain sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading. Both semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretcal findings are illustrated by numerical experiments.

Paper Structure

This paper contains 14 sections, 12 theorems, 86 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stability in time
  3. Generalization of Kopteva_Meng for any $r\ge 1$ (dropping the restriction $r\le\frac{2-\alpha}{\alpha}$)
  4. Error for a paradigm example without spatial derivatives
  5. Truncation error of the L1 fractional-derivative operator
  6. Truncation error for the semilinear term
  7. Pointwise-in-time error bound
  8. Error analysis for semilinear time-fractional parabolic equations
  9. Semidiscretization in time
  10. Maximum norm error for finite difference discretizations
  11. Finite element discretizations
  12. Quasilinear time-fractional subdiffusion equations
  13. Existence and uniqueness for \ref{['semi_semidiscr_method_quasi']} with $a_k=a(u)$
  14. Numerical results

Key Result

Theorem 2.2

Given $\gamma\in\mathbb{R}$ and $\lambda_0,\,\lambda_1\ge0$, let the temporal mesh satisfy $\lambda_0\tau_j^{\alpha}< \{\Gamma(2-\alpha)\}^{-1}$$\forall\,j\ge1$. Additionally, suppose that either one has t_grid_gen with some $r\ge 1$, or $\gamma\le \alpha-1$, and $\max\{\tau_j\}$ is sufficiently sma where $\ell_\gamma=\ell_\gamma(t_j):=1+\ln(t_j/\tau)$ for $\gamma=0$ and $\ell_\gamma:=1$ otherwise

Figures (3)

  • Figure 1: Test problem A: pointwise errors for \ref{['simplest_b']} with $\alpha = 0.3$, $\sigma = 2\alpha$, $M = 2^{17}$, $r=\frac{2-\alpha}{\sigma}$ (left) and $r=\frac{2-\alpha}{0.9\nu}$ (right), where $\nu:= 1+\sigma-\alpha$ .
  • Figure 2: Test problem A: pointwise errors for \ref{['simplest_b']} with $\alpha = 0.7$, $\sigma = \alpha/2$, $M = 2^{17}$, $r=\frac{2-\alpha}{\sigma}$ (left) and $r=\frac{2-\alpha}{0.9\nu}$ (right), where $\nu:= 1+\sigma-\alpha$.
  • Figure 3: Test problem B: maximum nodal errors for numerical scheme (\ref{['semi_semidiscr_method']} (first-order IMEX and second-order Newton-iteration IMEX versions) with $2^{10}$ degrees of freedom in space, $\sigma = \alpha/2$, $r = \frac{2-\alpha}{\sigma}$, $\alpha = 0.3$ (left) and $\alpha = 0.5$ (right).

Theorems & Definitions (32)

  • Remark 2.1
  • Theorem 2.2: stability in time
  • Remark 2.3: dropping the restriction $r\le\frac{2-\alpha}{\alpha}$
  • Lemma 2.4: comparison principle for the operator in \ref{['main_stab_new']}
  • Proof 1
  • Lemma 2.5: kopteva_semilin
  • Proof 2: Proof of Theorem \ref{['theo_main_stab_semi']}
  • Remark 2.6
  • Lemma 2.7
  • Proof 3
  • ...and 22 more