Nonconforming finite element methods on quadrilateral meshes

TL;DR

This paper introduces new families of nonconforming finite elements of any order on quadrilateral meshes, overcoming previous linear independence issues and generalizing lower order elements.

Contribution

It proposes two families of odd order and one family of even order nonconforming finite elements on quadrilaterals, with well-defined degrees of freedom and full shape function spaces.

Findings

Elements are well-defined for their shape function spaces.

Generalizes lower order nonconforming elements to any order.

Shows these spaces are full, a novel result in literature.

Abstract

It is well-known that it is comparatively difficult to design nonconforming finite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations. One reason lies in that these degrees of freedom associated to these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces, which explains why only several lower order nonconforming quadrilateral finite elements can be found in literature. The present paper proposes two families of nonconforming finite elements of any odd order and one family of nonconforming finite elements of any even order on quadrilateral meshes. Degrees of freedom are given for these elements, which are proved to be well-defined for their corresponding shape function spaces in a unifying way. These elements generalize three lower order nonconforming finite elements on quadrilaterals to any order. In addition, these nonconforming finite element spaces are shown to be full spaces which is somehow not discussed for nonconforming finite elements in literature before.