Volume 12, Issue 4
Conforming Hierarchical Basis Functions

M. J. Bluck

Commun. Comput. Phys., 12 (2012), pp. 1215-1256.

Published online: 2012-12

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  • 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 generalized 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. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair [16–18] and Arnold et al. [4] for simplices. The process of forming hierarchical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number. 

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@Article{CiCP-12-1215, author = {M. J. Bluck}, title = {Conforming Hierarchical Basis Functions}, journal = {Communications in Computational Physics}, year = {2012}, volume = {12}, number = {4}, pages = {1215--1256}, 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 generalized 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. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair [16–18] and Arnold et al. [4] for simplices. The process of forming hierarchical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number. 

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.090511.071211a}, url = {http://global-sci.org/intro/article_detail/cicp/7332.html} }
TY - JOUR T1 - Conforming Hierarchical Basis Functions AU - M. J. Bluck JO - Communications in Computational Physics VL - 4 SP - 1215 EP - 1256 PY - 2012 DA - 2012/12 SN - 12 DO - http://dor.org/10.4208/cicp.090511.071211a UR - https://global-sci.org/intro/cicp/7332.html KW - AB -

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 generalized 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. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair [16–18] and Arnold et al. [4] for simplices. The process of forming hierarchical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number. 

M. J. Bluck. (1970). Conforming Hierarchical Basis Functions. Communications in Computational Physics. 12 (4). 1215-1256. doi:10.4208/cicp.090511.071211a
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