Numerical analysis of nonlinear subdiffusion equations

TL;DR

This paper develops a comprehensive numerical analysis framework for nonlinear time-fractional parabolic equations, establishing error estimates and solution theory without extra regularity assumptions, validated by numerical experiments.

Contribution

It introduces a general fractional discrete Gr"onwall inequality framework and provides a complete solution theory for nonlinear fractional diffusion equations.

Findings

Error estimates of order O(h^2) and O(τ^α) for finite element and time discretization

Verification of the fractional Gr"onwall inequality for L1 scheme and BDF convolution quadrature

Numerical experiments confirming the sharpness of convergence rates

Abstract

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three technical tools: a fractional version of the discrete Gr\"onwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Gr\"onwall inequality, and verify it for the L1 scheme and convolution quadrature generated by BDFs. Further, we provide a complete solution theory, e.g., existence, uniqueness and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise $L^2(\Omega)$ norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order $O(h^2)$ (up to a logarithmic factor) and $O(\tau^\alpha)$, respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.