High accuracy analysis of mixed finite element method for nonlinear Klein-Gordon equation

This paper is concerned with a conforming mixed finite element approximation scheme for solving a class of nonlinear Klein-Gordon (KG) equations by using bilinear elements and Nedelec elements ($ Q_{11}+Q_{01}\times Q_{10} $). The existence and uniqueness of the approximate solution of the semi-discrete scheme is proved, and the superapproximation and superconvergence results are proved by constructing a mixed finite element projection. Furthermore, by adopting different processing techniques for nonlinear terms from previous literature, a linearized fully-discrete scheme is established, its superconvergence results are obtained, and finally numerical examples are used to demonstrate the correctness of the theoretical analysis and the effectiveness of the approximation scheme.