Conforming finite element function spaces in four dimensions, part I: Foundational principles and the tesseract

Nilima Nigam, David M. Williams

Research output: Contribution to journalArticlepeer-review

Abstract

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed ‘Finite Element Exterior Calculus’. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom.

Original languageEnglish (US)
Pages (from-to)198-223
Number of pages26
JournalComputers and Mathematics with Applications
Volume166
DOIs
StatePublished - Jul 15 2024

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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