Direct Serendipity and Mixed Finite Elements on Convex Polygons

DOI10.1007/S11075-022-01348-1arXiv2202.11229WikidataQ114224264 ScholiaQ114224264MaRDI QIDQ6391898

Todd Arbogast, Chuning Wang

Publication date: 22 February 2022

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.

Mathematics Subject Classification ID

Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical interpolation (65D05) de Rham theory in global analysis (58A12) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50)

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