2018/09/06 by Todd Arbogast, Zhen Tao, Arbogast, Todd +1
Computer Science · Engineering · #65D05 #65N12 #65N30 #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Structural Behavior of Reinforced Concrete
paper · pdf · doi:10.48550/arxiv.1809.02192
openalex publication_date 2018/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop 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 H1 or H(\textrmdiv) 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≥1, the new families of serendipity spaces \calDSr+1 are the precursors under the curl operator of our direct mixed finite element spaces Vr, which can be constructed to have full or reduced H(\textrmdiv) 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 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. Numerical results are presented to illustrate the properties of the new spaces.