Finite element differential forms on cubical meshes

Abstract

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.

v2: as accepted by Mathematics of Computation after minor revisions; v3: this version corresponds to the final version for Math. Comp., after copyediting and galley proofs

References in corpus (2)

• Finite element exterior calculus: from Hodge theory to numerical stability
• The serendipity family of finite elements

Cited by in corpus (14)

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• Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
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• Compatible finite element spaces for geophysical fluid dynamics
• Finite element differential forms on curvilinear cubic meshes and their approximation properties
• Spaces of finite element differential forms
• Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
• Construction of $H(\mathrm{div})$-Conforming Mixed Finite Elements on Cuboidal Hexahedra
• An Extended Galerkin analysis in finite element exterior calculus
• Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake
• A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus
• Constructions of some minimal finite element systems
• Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes