Direct serendipity and mixed finite elements on convex quadrilaterals

Abstract

AbstractThe classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and 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 $$H^1$$ H 1 or $$H(\text {div})$$ H ( div ) conformity, exactly two supplemental shape functions must be added to each element when $$r\ge 2$$ r ≥ 2 , and only one when $$r=1$$ r = 1 . 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 $$r\ge 1$$ r ≥ 1 , the new families of serendipity spaces $${\mathscr {DS}}_{r+1}$$ DS r + 1 are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full $$H(\text {div})$$ H ( div ) approximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. 10.1137/15M1013705). Other fully direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to fully direct mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties.