2012
DOI: 10.4208/cicp.090511.071211a
|View full text |Cite
|
Sign up to set email alerts
|

Conforming Hierarchical Basis Functions

Abstract: Abstract.A unified process for the construction of hierarchical conforming bases on a range of element types is proposed based on an ab initio preservation of the underlying cohomology. This process supports not only the most common simplicial element types, as are now well known, but is generalised to squares, hexahedra, prisms and importantly pyramids. Whilst these latter cases have received (to varying degrees) attention in the literature, their foundation is less well developed than for the simplicial case… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
2
1

Citation Types

1
6
0

Year Published

2013
2013
2024
2024

Publication Types

Select...
4
2
1

Relationship

0
7

Authors

Journals

citations
Cited by 8 publications
(7 citation statements)
references
References 31 publications
1
6
0
Order By: Relevance
“…First, the geometry: In this implementation, we describe the geometry using six noded Lagrangian triangular elements and/or nine noded quadratic square elements. Second, the surface fields must be approximated by basis functions: We will choose a class of hierarchical conforming basis functions (shown in Tables I and II up to third order) obtained as surface traces of the functions developed in [27]. These have been shown to be well-conditioned in a finite element sense.…”
Section: Differential Forms and The Efiementioning
confidence: 99%
See 1 more Smart Citation
“…First, the geometry: In this implementation, we describe the geometry using six noded Lagrangian triangular elements and/or nine noded quadratic square elements. Second, the surface fields must be approximated by basis functions: We will choose a class of hierarchical conforming basis functions (shown in Tables I and II up to third order) obtained as surface traces of the functions developed in [27]. These have been shown to be well-conditioned in a finite element sense.…”
Section: Differential Forms and The Efiementioning
confidence: 99%
“…These bases are detailed in [27]. In Section III, a p-MUS multilevel preconditioner is described, together with results demonstrating its effectiveness on selected problems.…”
Section: Introductionmentioning
confidence: 99%
“…Verification output Associated routine in [38] Section V-A, [19], [20] Hierarchical, subspace decomposition From third-order, not in R k check_webb_ap99.m check_sun_jsc01.m Section V-B, [30] Full-order, subspace decomposition One family of face functions not in S k check_beuchler_aam13.m Section V-B, [28] Full-order, subspace decomposition Rotational functions not in S k check_xin_jcm11.m Section V-C, [29] Unified process Accuracy problems for the nullspace, not in R k check_bluck_ccp12.m Section V-D, [17], [27] High-order Whitney functions All functions in R k check_graglia_ap11.m check_fuentes_camwa15.m Section V-E, [23] Hierarchical, subspace decomposition All functions in R k check_ingelstrom_mtt06.m TABLE II: Overview of Section V with routines included in [38].…”
Section: Propertiesmentioning
confidence: 99%
“…C. Curl-conforming and div-conforming hierarchical functions: [29] This work presents a unified process for the construction of curl and div-conforming functions for the whole range of shapes (simplices, squares, hexahedra, prisms and pyramids). The expressions are presented in the paper ready to use, presenting mixed and full-order functions (the so-called B and C classes following a classification suggested in [27]).…”
Section: Propertiesmentioning
confidence: 99%
“…From the perspective of differential forms Hipmair [13] gave a canonical construction of the H(curl)and H(div)-conforming R n simplicial elements. See also the related works [14][15][16][17] from such a perspective. In addition to other results Ainsworth and Coyle [18] constructed hierarchical bases of arbitrary order for H(div)-conforming tetrahedral element.…”
Section: Introductionmentioning
confidence: 99%