Direct serendipity and mixed finite elements on convex quadrilaterals

Abstract

The 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.

Direct serendipity and mixed finite elements on convex polygons

“…Moreover, it follows from the construction that we obtain global H 1 conforming elements by just matching vertex and edge DoFs on the boundaries of the elements; that is, local basis functions merge together continuously to give a global nodal basis for DS r = DS r ( ) ⊂ H 1 ( ). Our construction directly extends that given in [5] for the case N = 4.…”

“…The proof follows closely that given in [5] for the quadrilateral case and so is omitted here except for discussion of one important issue. The proof uses a continuous dependence argument, relying on the fact that the set of vertices lies in a compact set as well as Assumption 5.1, which ensures that the construction of the finite elements on E N depends continuously on its vertices.…”

“…On triangular and rectangular elements when r ≥ 1, it is known that the classic serendipity space S r+1 (in place of DS r+1 above) is the precursor of the Brezzi-Douglas-Marini mixed finite element space BDM r [1,3,4] (in place of V r−1 r above). It is also known that on quadrilateral elements, the direct serendipity space is the precursor of the direct mixed spaces [5]. The families of mixed finite elements on E N , N > 4, are new.…”

“…Recently, the current authors and Z. Tao [5] constructed serendipity spaces directly on quadrilaterals of any approximation order r +1 ≥ 2 without using a mapping from a reference element. The resulting new family of spaces were called direct serendipity finite elements and denoted DS r (E), r ≥ 1.…”

“…The method works for any order of accuracy r, but it uses even more interior DoFs, 1 2 r(r − 1). In the rest of this paper, we generalize the construction in [5] to a general strictly convex polygon E N with N vertices. Our construction does not extend to non strictly convex polygons.…”