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

作者：Shi Dongyang^*; Pei Lifang

来源：Applied Mathematics and Computation, 2013, 219(17): 9447-9460.

DOI：10.1016/j.amc.2013.03.008

• Nonlinear sine-Gordon equation
• Quadrilateral meshes
• Quasi-Wilson element
• Optimal order error estimate
• Fully-discrete
• Superconvergence

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(h(2)) (h denotes the mesh size) in H-1-norm, which leads to optimal order error estimate and superclose result with order O(h(2)) 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(h(3)) 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 H-1-norm is presented on generalized rectangular meshes by anew technique. Thirdly, the global superconvergence results of H-1-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.

• 出版日期2013-5-1
• 单位郑州大学; 洛阳理工学院