Mathematical Analysis for Quadrilateral Rotated $Q_1$ Element III: The Effect of Numerical Integration

Abstract

This is the third part of the paper for the rotated $Q_1$ nonconforming element on quadrilateral meshes for general second order elliptic problems. Some optimal numerical formulas are presented and analyzed. The novelty is that it includes a formula with only two sampling points which excludes even a $Q_1$ unisolvent set. It is the optimal numerical integration formula over a quadrilateral mesh with least sampling points up to now.