Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations

Nonconforming quadrilateral finite element method (FEM) of the two-dimensional nonlinear sine-Gordon equation is studied for semi-discrete and Crank-Nicolson fully-discrete schemes, respectively. Firstly, we prove a special feature of a new arbitrary quadrilateral element (named modified Quasi-Wilson element), i.e., the consistency error is of order O(h2) (h denotes the mesh size) in H1-norm, which leads to optimal order error estimate and superclose result with order O(h2) for the semi-discrete scheme through a different approach from the existing literature. Secondly, because the consistency error estimate of the new modified Quasi-Wilson element can reach a staggering O(h3) order, two orders higher than that of interpolation error, the optimal order error estimates of Crank-Nicolson fully-discrete scheme are obtained on arbitrary quadrilateral meshes with Ritz projection. Moreover, a superclose result in H1-norm is presented on generalized rectangular meshes by a new technique. Thirdly, the global superconvergence results of H1-norm for both semi-discrete and fully-discrete schemes are derived on rectangular meshes with interpolated postprocessing technique. Finally, a numerical test is carried out to verify the theoretical analysis.