Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems (Communications in Computational Physics, 2018)
This paper is concerned with numerical solutions of time-fractional nonlinearparabolic problems by a class of L1-Galerkin finite element methods. The analysisof L1 methods for time-fractional nonlinear problems is limited mainly due to the lackof a fundamental Gronwall type inequality. In this paper, we establish such a fundamentalinequality for the L1 approximation to the Caputo fractional derivative. Interms of the Gronwall type inequality, we provide optimal error estimates of severalfully discrete linearized Galerkin finite element methods for nonlinear problems. Thetheoretical results are illustrated by applying our proposed methods to the time fractionalnonlinear Huxley equation and time fractional Fisher equation.
Keywords: Time-fractional nonlinear parabolic problems; L1-Galerkin FEMs; Error estimates; discrete fractional Gronwall type inequality; Linearized schemes.