  We construct new families of direct serendipity and direct mixed finite
elements on general planer convex polygons that are $H^1$ and $H(div)$
conforming, respectively, and possess optimal order of accuracy for any order.
They have a minimal number of degrees of freedom subject to the conformity and
accuracy constraints. The name arises because the shape functions are defined
directly on the physical elements, i.e., without using a mapping from a
reference element. The finite element shape functions are defined to be the
full spaces of scalar or vector polynomials plus a space of supplemental
functions. The direct serendipity elements are the precursors of the direct
mixed elements in a de Rham complex. The convergence properties of the finite
elements are shown under a regularity assumption on the shapes of the polygons
in the mesh, as well as some mild restrictions on the choices one can make in
the construction of the supplemental functions. Numerical experiments on
various meshes exhibit the performance of these new families of finite
elements.
