Constructions of some minimal finite element systems (Q2814660)

Introducing the main concepts of finite element systems (FES), this study is focused on minimal compatible FES (mcFES) which has three key properties that it is compatible, it contains certain prescribed functions, and it has the smallest dimension among all possible FES with these properties. Following the results announced without proofs by the first author [C. R., Math., Acad. Sci. Paris 348, No. 3--4, 217--221 (2010; Zbl 1186.65148)], it is shown here how the dimension of an mcFES can be computed in terms of certain cohomology groups and how an mcFES that contains a given set of functions can be constructed, within a larger compatible FES. This analysis and construction process is applied to the trimmed polynomial spaces defined by \textit{R. Hiptmair} [Math. Comput. 68, No. 228, 1325--1346 (1999; Zbl 0938.65132)], the serendipity spaces defined by \textit{D. N. Arnold} and \textit{G. Awanou} [Math. Comput. 83, No. 288, 1551--1570 (2014; Zbl 1297.65142)] and the TNT elements, defined by \textit{B. Cockburn} and \textit{W. Qiu} [Math. Comput. 83, No. 286, 603--633 (2014; Zbl 1290.65109)]. A dimension formula for mcFES containing polynomials of a given degree on hypercubes is also presented here.