A class of nonparametric DSSY nonconforming quadrilateral elements

TL;DR

This paper introduces a new class of nonparametric nonconforming quadrilateral finite elements that require fewer degrees of freedom and maintain optimal convergence, improving upon existing parametric DSSY elements.

Contribution

The paper develops a nonparametric DSSY quadrilateral element with only four degrees of freedom, simplifying the design while preserving convergence properties.

Findings

Requires only four degrees of freedom per element.

Achieves optimal order of convergence.

Numerical results confirm effectiveness and comparison with parametric DSSY elements.

Abstract

A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J.Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming {G}alerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM--Math. Model. Numer. Anal., 33(4):747--770, 1999]. The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming quadrilateral finite elements: A correction. Calcolo, 37(4):253--254, 2000], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.