Commun. Comput. Phys., 14 (2013), pp. 621-638.


On the Construction of Well-Conditioned Hierarchical Bases for H(div)-Conforming R^n Simplicial Elements

Jianguo Xin 1, Wei Cai 1*, Nailong Guo 2

1 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA.
2 Mathematics and Computer Science Department, Benedict College, Columbia, SC 29204, USA.

Received 10 April 2012; Accepted (in revised version) 4 November 2012
Available online 25 January 2013
doi:10.4208/cicp.100412.041112a

Abstract

Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universitat, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.

AMS subject classifications: 65N30, 65F35, 65F15

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Key words: Hierarchical bases, simplicial H(div)-conforming elements, matrix conditioning.

*Corresponding author.
Email: wcai@uncc.edu (W. Cai)
 

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