The Numerical Implementation Of Quadratic Nonconforming Finite Element Method Over Arbitrary Quadrilateral Meshes

The finite element method is a very efficient numerical method for the partial differential equations, which is widely used in scientific computing and engineering fields. The noncon-forming finite element methods successfully provide stable numerical solutions for many practical fluid flow and solid mechanics problems, for instance, Stokes, Navier-Stokes problems and elasticity related problems. Therefore, the nonconforming finite element methods draw wide attention from scientists and engineers. Even though the triangular or tetrahedral meshes are popular to use, in some cases where the geometry of the problem has a quadrilateral nature, one wishes to use quadrilateral or hexahedral meshes with proper elements.A numerical method includes its mathematical foundation and its implementation, while researching theory foundation, the realization of the methods is also important. This paper is based on a new quadratic nonconforming finite element on quadrilaterals, which is continuous at the two Gauss points on each edge of the quadrilateral and is proposed by Kim, Luo and Sheen, et al. The aim of this paper is using this finite element to solve the second-order elliptic problems in two dimensions. In the solving process, the element stiffness matrix and load vector are represented by integral, so the structure of the corresponding numerical integral formulas is an important issue to solve in this paper. First, we change the boundary value problem of the elliptic equation into its variational problem, construct the corresponding element stiffness matrix and give its calculating methods. Then we calculate the load vector. Considering the quadratic nonconforming finite element only continuous at the eight Gauss points, we construct a new quadratic integral formula on quadrilaterals, which is accurate for the piecewise quadratic polynomial continuous at the eight Gauss points, especially on a rectangular element the integral formula has positive weight coefficient with a degree of accuracy3. Finally, some numerical experiments are given to prove the convergence of the quadratic nonconforming finite element method.