Direct Serendipity and Mixed Finite Elements on Convex Quadrilaterals

TL;DR

This paper introduces direct serendipity and mixed finite element spaces on convex quadrilaterals that achieve optimal approximation with minimal local dimension, directly defined on elements without reference mappings.

Contribution

The paper develops new direct finite element spaces that are optimal, minimal, and directly defined on convex quadrilaterals, improving approximation properties over classical methods.

Findings

Achieve optimal approximation with minimal local dimension.

Construct spaces directly on elements without reference mappings.

Numerical results confirm the effectiveness of the new spaces.

Abstract

The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop $\textit{direct serendipity}$ and $\textit{direct mixed}$ finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree $r$, respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global $H^1$ or $H(\textrm{div})$ conformity, exactly two supplemental shape functions must be added to each element. The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index $r\ge1$, the new families of serendipity spaces ${\cal{DS}}_{r+1}$ are the precursors under the curl operator of our direct mixed finite element spaces ${\mathbf{V}}_r$, which can be constructed to have full or reduced $H(\textrm{div})$ approximation properties. One choice of direct serendipity supplements is the precursor of the recently introduced Arbogast-Correa spaces [SIAM J. Numer. Anal., 54 (2016), pp.~3332--3356]. Other $\textit{fully}$ direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to $\textit{fully}$ direct mixed spaces. Numerical results are presented to illustrate the properties of the new spaces.