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Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

Natalia Kopteva

TL;DR

This work analyzes a semilinear subdiffusion problem with Caputo time derivative by extending sharp pointwise-in-time error bounds from linear to semilinear cases using a three-point L2-type time discretization on graded meshes. The method achieves order $3-\alpha$ in time and leverages an inverse-monotone property of the discrete operator $\delta_t^{\alpha}-\lambda$ to establish a discrete comparison principle, enabling stable, accurate handling of the Lipschitz nonlinearity. The main contributions are (i) proving inverse-monotonicity for $\delta_t^{\alpha}-\lambda$ under mild mesh assumptions, (ii) deriving explicit stability and error bounds $\|u(\cdot,t_m)-U^m\|_{L_2(\Omega)} \lesssim \mathcal E^m$ that depend on the graded mesh parameter $r$, and (iii) providing a complete error analysis for semidiscrete schemes on general meshes satisfying A2 and A3, including the nonlinear truncation error. These results enable sharp, practical error control for semilinear subdiffusion models in applications where the solution exhibits initial-time singularities.

Abstract

A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order $3-α$, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

TL;DR

This work analyzes a semilinear subdiffusion problem with Caputo time derivative by extending sharp pointwise-in-time error bounds from linear to semilinear cases using a three-point L2-type time discretization on graded meshes. The method achieves order in time and leverages an inverse-monotone property of the discrete operator to establish a discrete comparison principle, enabling stable, accurate handling of the Lipschitz nonlinearity. The main contributions are (i) proving inverse-monotonicity for under mild mesh assumptions, (ii) deriving explicit stability and error bounds that depend on the graded mesh parameter , and (iii) providing a complete error analysis for semidiscrete schemes on general meshes satisfying A2 and A3, including the nonlinear truncation error. These results enable sharp, practical error control for semilinear subdiffusion models in applications where the solution exhibits initial-time singularities.

Abstract

A semilinear initial-boundary value problem with a Caputo time derivative of fractional order is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order , we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.
Paper Structure (3 sections, 6 theorems, 14 equations)

This paper contains 3 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction. Main result
  2. Discrete comparison principle and stability properties for $\delta_t^\alpha-\lambda$
  3. Error analysis. Proof of Theorem \ref{['theo_semi']}

Key Result

Theorem 1.2

Let the graded temporal mesh $\{t_j=T(j/M)^r\}_{j=0}^M$, with any fixed $r\ge1$, satisfy $\lambda\tau_j^{\alpha}< \{\Gamma(2-\alpha)\}^{-1}$$\forall\,j\ge1$ Suppose that $u$ is a unique solution of problem with the initial data $u_0\in L_\infty(\Omega)$ and under assumption A1 on $f$. Also, for $t\

Theorems & Definitions (9)

  • Theorem 1.2
  • Lemma 2.2: Comparison principle for $\delta_t^\alpha-\lambda$
  • proof
  • Theorem 2.3: Stability of $\delta_t^\alpha-\lambda$
  • Corollary 2.4
  • Lemma 3.1: Stability for parabolic case
  • proof
  • Theorem \ref{theo_semi}${}^*$
  • proof