Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems
Provides a key theoretical tool (Gronwall inequality) for analyzing L1 methods in time-fractional nonlinear problems, addressing a known bottleneck in numerical analysis.
This paper establishes a fundamental Gronwall-type inequality for the L1 approximation of the Caputo fractional derivative, enabling optimal error estimates for L1-Galerkin FEMs applied to time-fractional nonlinear parabolic problems. Numerical examples confirm the theoretical results.
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.