Nonconforming Rational Finite Elements Of Higher Order On Arbitrary Convex Quadrilateral Meshes

The finite element method is developed based on the variational principle. It is widely used in scientific computing and engineering fields as a numerical analysis method. The nonconform-ing finite element methods successfully provide stable numerical solutions for the practical fluid flow and solid mechanics problems comparing with conforming finite element. So they draw in-creasing attention from more scientists lately. Recently, Lee and Sheen constructed a quadratic nonconforming finite element on rectangles, Meng and Luo constructed a cubic nonconforming finite element on rectangles. But the construction method can not be used on the arbitrary quadrilaterals directly. So how to construct the nonconforming finite element on the arbitrary quadrilateral becomes a problem to solve. Some scholars introduce nonconforming finite element on the arbitrary convex quadrilaterals with accelerating the development of the finite element in recent years.This paper introduces one nonconforming finite element of higher order especially second order and that of third order on genuinely quadrilateral meshes. The degrees of freedom(DOFs) are defined by the values at the two(or three for third-order case) Gauss-Legendre points on every edge of the quadrilateral. For second-order case, one more interior DOF is required and we choose the value of intersection of diagonal in this paper. These two finite element spaces consist of some rational functions plus P2 and P3. Due to the existence of one linear relation among the above DOFs. it turns out the numbers of DOFs are locally eight for second-order and eleven for third-order case respectively. Global basis functions are defined and the corresponding dimensions are counted for the Dirichlet boundary value problem of second-order elliptic-problem. These1 two finite element spaces are quadratic-and cubic on the perimeter of the quadrilateral and hence error estimates of optimal order are easily obtained in both broken energy and L2 norms. Finally, numerical examples are provided and match our theoretical results very well.