A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equations

Abstract

In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered.