TY - JOUR
T1 - Conforming finite element function spaces in four dimensions, part I
T2 - Foundational principles and the tesseract
AU - Nigam, Nilima
AU - Williams, David M.
N1 - Publisher Copyright:
© 2024 Elsevier Ltd
PY - 2024/7/15
Y1 - 2024/7/15
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85193505869&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85193505869&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2024.05.005
DO - 10.1016/j.camwa.2024.05.005
M3 - Article
AN - SCOPUS:85193505869
SN - 0898-1221
VL - 166
SP - 198
EP - 223
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
ER -