Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion

TL;DR

This work develops a rigorous framework for a quasilinear subdiffusion model with Caputo time derivative $\partial_t^α$ and nonlinear diffusion $a(u)$. It establishes new pointwise-in-time regularity results and a Hölder-type perturbation estimate, then proposes a corrected second-order backward differentiation formula-based convolution-quadrature scheme to discretize time. The authors prove a nearly optimal $O(τ^{1+α-ε})$ error bound in $L^2(Ω)$ without extra regularity assumptions, using a linear-nonlinear splitting and detailed operator estimates, and validate the theory with 2D numerical experiments showing sharp convergence behavior. Together, the results enable robust, high-order time stepping for quasilinear subdiffusion with low regularity data and guide future extensions to other subdiffusion formulations.

Abstract

In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order $α\in (0,1)$ in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several pointwise-in-time regularity estimates that are useful for numerical analysis. Then we develop a high-order time stepping scheme for solving quasilinear subdiffusion, based on convolution quadrature generated by second-order backward differentiation formula with correction at the first step. Further, we establish that the convergence order of the scheme is $O(τ^{1+α-ε})$ without imposing any additional assumption on the regularity of the solution. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions show the sharpness of the error estimate.

Figures (1)

• Figure 1: Initial condition, solutions at different $t$ and stationary solution $u^*$.

Theorems & Definitions (30)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof