Direct Serendipity and Mixed Finite Elements on Convex Polygons

TL;DR

This paper introduces new direct serendipity and mixed finite element families on convex polygons that are conforming, optimal in accuracy, and have minimal degrees of freedom, with proven convergence and demonstrated numerical performance.

Contribution

It develops the first direct finite element families on convex polygons that are conforming, optimal, and minimal in degrees of freedom, without reference mappings.

Findings

Finite elements are optimal in accuracy for any order.

Numerical experiments confirm the theoretical convergence.

Elements have minimal degrees of freedom for given constraints.

Abstract

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.