Numerical Implementation Of A Piecewise P₂-Nonconforming Quadrilateral Finite Element

In the practical engineering calculation,many complex partial differential equations will be generated.The numerical method we mainly using to solve this kind of complex partial differential equations is the finite element method.While we construct the finite element space,it is equally important to give efficient and convenient calculation methods.In 2013,Kim,Luo and Meng constructed a piecewise quadratic nonconforming finite element on an arbitrary convex quadrilateral by using spline function space.In this paper,the piecewise quadratic spline element is used to construct a numerical method for solving second-order elliptic problems.Because the finite element can not adopt a unified numerical integration scheme on the element,the basis function of the finite element is re-expressed by using the B-net basis on the triangle.Since Bernstein polynomials have affine invariance and integrals can be expressed explicitly,we express the basis functions in the form of B-nets,The derivation and integration of the finite element basis functions can be transformed into the linear combination of the vectors or matrices composed of their B-net coefficients,so that the stiffness matrix can be displayed.In addition,because the degrees of freedom we choosing are Gauss points on the four sides of the quadrilateral,when assembling the element stiffness matrix,we can transform the integration into the form of linear combination of degrees of freedom,which saves the time of numerical integration.The final numerical results show that our method can significantly improve the calculation speed.