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Volume 29, Issue 5
On the Construction of Well-Conditioned Hierarchical Bases for Tetrahedral $\mathcal{H}{(curl)}$-Conforming Nédélec Elements

Jianguo Xin, Nailong Guo, & Wei Cai

J. Comp. Math., 29 (2011), pp. 526-542.

Published online: 2011-10

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  • Abstract

A partially orthonormal basis is constructed with better conditioning properties for tetrahedral $\mathcal{H}{(curl)}$-conforming Nédélec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonality. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi-stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58:2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix $µM+S$ has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter $µ$.

  • AMS Subject Headings

65N30, 65F35, 65F15.

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COPYRIGHT: © Global Science Press

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@Article{JCM-29-526, author = {}, title = {On the Construction of Well-Conditioned Hierarchical Bases for Tetrahedral $\mathcal{H}{(curl)}$-Conforming Nédélec Elements}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {5}, pages = {526--542}, abstract = {

A partially orthonormal basis is constructed with better conditioning properties for tetrahedral $\mathcal{H}{(curl)}$-conforming Nédélec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonality. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi-stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58:2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix $µM+S$ has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter $µ$.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1103-m3464}, url = {http://global-sci.org/intro/article_detail/jcm/10383.html} }
TY - JOUR T1 - On the Construction of Well-Conditioned Hierarchical Bases for Tetrahedral $\mathcal{H}{(curl)}$-Conforming Nédélec Elements JO - Journal of Computational Mathematics VL - 5 SP - 526 EP - 542 PY - 2011 DA - 2011/10 SN - 29 DO - http://doi.org/10.4208/jcm.1103-m3464 UR - https://global-sci.org/intro/article_detail/jcm/10383.html KW - Hierarchical bases, Tetrahedral $\mathcal{H}{(curl)}$-conforming elements, Matrix conditioning. AB -

A partially orthonormal basis is constructed with better conditioning properties for tetrahedral $\mathcal{H}{(curl)}$-conforming Nédélec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonality. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi-stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58:2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix $µM+S$ has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter $µ$.

Jianguo Xin, Nailong Guo, & Wei Cai. (1970). On the Construction of Well-Conditioned Hierarchical Bases for Tetrahedral $\mathcal{H}{(curl)}$-Conforming Nédélec Elements. Journal of Computational Mathematics. 29 (5). 526-542. doi:10.4208/jcm.1103-m3464
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