Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems

TL;DR

This paper develops a fundamental Gronwall type inequality for $L1$-Galerkin finite element methods, enabling optimal error estimates for solving time-fractional nonlinear parabolic problems, demonstrated through three example equations.

Contribution

It introduces a new Gronwall inequality for $L1$ methods, facilitating rigorous error analysis for nonlinear time-fractional PDEs.

Findings

Established a fundamental Gronwall type inequality for $L1$ approximation.

Provided optimal error estimates for fully discrete linearized Galerkin methods.

Validated the theoretical results with numerical examples on three equations.

Abstract

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of $L1$-Galerkin finite element methods. The analysis of $L1$ methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the $L1$ approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to three examples: linear Fokker-Planck equation, nonlinear Huxley equation and Fisher equation.