Unconditional convergence analysis of two linearized Galerkin FEMs for the nonlinear time-fractional diffusion-wave equation

Abstract

In this paper, we present and analyze two linearized Galerkin finite element schemes, which are constructed by employing the H2N2 formula and its fast version in time direction, for solving the nonlinear time-fractional diffusion-wave equation. By utilizing mathematical induction, the optimal error estimates in H1-norm are derived without any ratio restrictions between the time step size τ and the space mesh size h. The key point in our argument is the application of Sobolev’s embedding inequality to the fully discrete solution uhn. On the other hand, additional time-discrete elliptic system and the inverse inequality, which play a vital role in the temporal–spatial error splitting technique, are avoided in our numerical analysis. Finally, two numerical experiments are given to demonstrate the theoretical findings.