Simple quadrature rules for a nonparametric nonconforming quadrilateral element

TL;DR

This paper presents simple, efficient quadrature rules for nonparametric nonconforming quadrilateral finite elements, improving computational efficiency and convergence for numerical solutions in finite element analysis.

Contribution

Introduction of new two- and three-point quadrature rules for nonconforming quadrilateral elements that are asymptotically optimal and easy to implement.

Findings

Quadrature rules are asymptotically optimal for small mesh sizes.

Numerical experiments confirm efficiency and convergence.

New rules simplify computations for nonconforming quadrilateral elements.

Abstract

We introduce simple quadrature rules for the family of nonparametric nonconforming quadrilateral element with four degrees of freedom. Our quadrature rules are motivated by the work of Meng {\it et al.} \cite{meng2018new}. First, we introduce a family of MVP (Mean Value Property)-preserving four DOFs nonconforming elements on the intermediate reference domain introduced by Meng {\it et al.}. Then we design two--points and three--points quadrature rules on the intermediate reference domain. Under the assumption on equal quadrature weights, the deviation from the quadrilateral center of the Gauss points for the two points and three points rules assumes the same quadratic polynomials with constant terms modified. Thus, the two--points rule and three--points rule are constructed at one stroke. The quadrature rules are asymptotically optimal as the mesh size is sufficiently small. Several numerical experiments are carried out, which show efficiency and convergence properties of the new quadrature rules.