Numerical Implementation Of An 8-FOF Non Parametric Element On Arbitrary Quadrilateral

As a tool for solving differential equations,finite element method has been widely used and developed in recent years.When solving the second order elliptic equation,the conforming finite element space requires the shape function to be continuous in the whole solution region,while the nonconforming element cancels the continuity requirement and reduces the construction parameters accordingly.In the two-dimensional case,triangle or quadrilateral is usually used to divide the solution region,although the technology of triangle and tetrahedron has become mature and has good flexibility,when the geometry of the problem has a quadrilateral nature or requires less degrees of freedom,quadrilateral subdivision has more advantages.Therefore,it is of practical significance to construct nonconforming elements on any quadrilateral.In 2016,Meng and Zhou proposed a nonconforming element on an arbitrary convex quadrilateral.A linear constraint is added to the constructed degrees of freedom,so the dimension of the corresponding space is reduced by 1.Because of this linear constraint,we have to solve a system of linear equations in each element.In addition,the element has higher order bubble function and requires numerical integration on any quadrilateral.These two factors lead to huge computation.In this paper,the above finite element calculation is simplified from two perspectives.In the first method,we can map the values of the reference element to any element by using the affine transformation.Moreover,the scale relation on the original line can be kept unchanged after mapping.By this method,we can express the coefficients of the shape function on each element in a unified format,which saves the time of solving linear equations on each element.In addition,because the selected degrees of freedom of the element is zero-order moments,first-order moments on any quadrilateral boundary and the integral mean values on the element,in the second method,we use the partial integration formula to transform the calculation of the stiffness matrix into the form of linear combination of the degrees of freedom of the shape functions,and express the calculation process of the stiffness matrix by the product of the matrix and the vector.Compared with the general method,we need to use 45 times of numerical integration on each element,and our method only needs 6 times of numerical integration on each element.Finally,the two methods are evaluated by solving the second order elliptic equation,and the results show the advantages of the two methods.