Unconditional superconvergence analysis of a two-grid finite element method for nonlinear wave equations

Abstract

This paper aims to present a two-grid method (TGM) with low order nonconforming EQ1rot finite element for solving a class of nonlinear wave equations, and to give the superconvergent error analysis unconditionally. Firstly, for the Crank-Nicolson fully discrete scheme, the existence and uniqueness of the numerical solutions are proved, and based on the special characters of this element as well as the priori estimates, the supercloseness and superconvergence analysis of both the original variable u and the auxiliary variable q=ut in the broken H1-norm are deduced on the coarse mesh for the Galerkin finite element method (FEM). Then, by employing the interpolation postprocessing approach and the boundness of the numerical solution in the broken H1-norm on the coarse mesh, the corresponding superconvergence results of order O(τ2+H4+h2) for the TGM are obtained unconditionally, here τ, H and h denote time step, coarse and fine grid sizes, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.