Effect of numerical integration on a new rotated nonconforming quadrilateral element

A new nonparametric nonconforming quadrilateral finite element is used to approximate the general second-order elliptic problem in two dimensions. Some optimal numerical integration formulas are presented and analyzed. These formulas are derived on a reference quadrilateral which can be linearly mapped to a physical quadrilateral. The novelty of these formulas is that they only involve two quadrature nodes which excludes even a Q(1) unisolvent set, and they are not required to be exact for all the shape functions. Numerical tests show that the presented quadrature rules can also be used coupled with other low-order elements. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

A new nonparametric nonconforming quadrilateral finite element is introduced to approximate the general second-order elliptic problem in two dimensions, along with optimal numerical integration formulas. These formulas, derived on a reference quadrilateral and involving only two quadrature nodes, are not required to be exact for all shape functions and can be used with other low-order elements in numerical tests.