Direct serendipity and mixed finite elements on convex polygons

• Published in: Numerical algorithms Vol. 92; no. 2; pp. 1451 - 1483
• Main Authors: Arbogast, Todd, Wang, Chuning
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2023; Springer Nature B.V
• Subjects: Accuracy; Algebra; Algorithms; Approximation; Computer Science; Construction; Mathematical analysis; Numeric Computing; Numerical Analysis; Original Paper; Polygons; Polynomials; Serendipity; Shape functions; Theory of Computation; Direct finite elements; Serendipity finite elements; 65N12; Finite element exterior calculus; Polygonal meshes; Generalized barycentric coordinates; 65D05; Optimal approximation; 65N30
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-022-01348-1

We construct new families of direct serendipity and direct mixed finite elements on general planar, strictly 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.