ICES REPORT 17-28 October 2017 Direct Serendipity Finite Elements on Convex Quadrilaterals

The classical serendipity finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop the direct serendipity spacesDSr, a new family of finite elements for r ≥ 2 that has the same number of degrees of freedom as the classical space but maintains optimal approximation properties. The set of local shape functions forDSr contains the full set of polynomials of degree r defined directly on each element. Because there are not enough degrees of freedom, exactly two supplemental rational functions are added to each element.