  The classical serendipity and mixed finite element spaces suffer from poor
approximation on nondegenerate, convex quadrilaterals. In this paper, we
develop $\textit{direct serendipity}$ and $\textit{direct mixed}$ 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
$H^1$ or $H(\textrm{div})$ 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 $r\ge1$, the new families of
serendipity spaces ${\cal{DS}}_{r+1}$ are the precursors under the curl
operator of our direct mixed finite element spaces ${\mathbf{V}}_r$, which can
be constructed to have full or reduced $H(\textrm{div})$ 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 $\textit{fully}$ direct serendipity supplements
can be defined without the use of mappings from reference elements, and these
give rise in turn to $\textit{fully}$ direct mixed spaces. Numerical results
are presented to illustrate the properties of the new spaces.
