Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T17:27:40.988Z Has data issue: false hasContentIssue false
  • English
  • Français

A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements

Published online by Cambridge University Press:  20 August 2015

Jianguo Xin  and
Wei Cai
Jianguo Xin*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223, USA
Wei Cai*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

We construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

Keywords

65N3065F3565F15Hierarchical basesH(curl)-conforming elementsmatrix conditioningclassic orthogonal polynomials
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 780 - 806
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abdul-Rahman, R. and Kasper, M., Orthogonal hierarchical Nédélec elements, IEEE Trans. Magn., 44 (2008), 1210–1213.CrossRefGoogle Scholar
[2]Adjerid, S., Aiffa, M. and Flaherty, J. E., Hierarchical finite element bases for triangular and tetrahedral elements, Comput. Methods. Appl. Mech. Engrg., 190 (2001), 2925–2941.CrossRefGoogle Scholar
[3]Ainsworth, M. and Coyle, J., Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes, Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709–6733.CrossRefGoogle Scholar
[4]Ainsworth, M. and Coyle, J., Conditioning of hierarchic p-version Nédélec elements on meshes of curvilinear quadrilaterals and hexahedra, SIAM J. Numer. Anal., 41 (2003), 731–750.Google Scholar
[5]Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numer. Methods. Engrg., 58 (2003), 2103–2130.Google Scholar
[6]Babžka, I., Szabo, B. A. and Katz, I. N., The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515–545.Google Scholar
[7]Bittencourt, M. L., Fully tensorial nodal and modal shape functions for triangles and tetra-hedra, Int. J. Numer. Methods. Engrg., 63 (2005), 1530–1558.Google Scholar
[8]Bossavit, A., Computational Electromagnetism, Academic Press, New York, 1998.Google Scholar
[9]Dunkl, C. F. and Xu, Y., Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications, 81, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
[10]Zhang, F. (Editor), The Schur Complement and Its Applications, Springer-Verlag, New York, 2005.Google Scholar
[11]Gopalakrishnan, J., García-Castillo, L. E. and Demkowicz, L. F., Nédélec spaces in affine coordinates, Comput. Math. Appl., 49 (2005), 1285–1294.Google Scholar
[12]Hiptmair, R., Canonical construction of finite elements, Math. Comput., 68 (1999), 1325–1346.Google Scholar
[13]Hiptmair, R., Higher order Whitney forms, PIER., 32 (2001), 271–299.Google Scholar
[14]Ingelström, P., A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes, IEEE Trans. Microw. Theor. Tech., 54 (2006), 106–114.Google Scholar
[15]Jørgensen, E., Volakis, J. L., Meincke, P. M. and Breinbjerg, O., Higher order hierarchical Legen-dre basis functions for electromagnetic modeling, IEEE Trans. Antenna. Propaga., 52 (2004), 2985–2995.Google Scholar
[16]Karniadakis, G. E. and Sherwin, S. J., Spectral/hp Element Methods for CFD, Oxford University Press, New York, 1999.Google Scholar
[17]Nédélec, J. C., Mixed finite elements in R3, Numer. Math., 35 (1980), 315–341.Google Scholar
[18]Rachowicz, W. and Demkowicz, L., An hp-adaptive finite element method for electromagnetics, II, a 3D implementation, Int. J. Numer. Methods. Engrg., 53 (2002), 147–180.Google Scholar
[19]Rapetti, F., High order edge elements on simplicial meshes, ESAIM: M2AN, 41 (2007), 1001–1020.Google Scholar
[20]Rapetti, F. and Bossavit, A., Whitney forms of higher degree, SIAM J. Numer. Anal., 47 (2009), 2369–2386.Google Scholar
[21]Ren, Z. and Ida, N., High order differential form-based elements for the computation of electromagnetic field, IEEE Trans. Magn., 36 (2000), 1472–1478.Google Scholar
[22]Schöberl, J. and Zaglmayr, S., High order Nédélec elements with local complete sequence properties, Compel., 24 (2005), 374–384.Google Scholar
[23]Shen, J., Efficient spectral-Galerkin method, I, direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489–1505.CrossRefGoogle Scholar
[24]Shreshevskii, I. A., Orthogonalization of graded sets of vectors, J. Nonlinear. Math. Phys., 8 (2001), 54–58.CrossRefGoogle Scholar
[25]Sun, D.-K., Lee, J.-F. and Cendes, Z., Construction of nearly orthogonal Nedelec bases for rapid convergence with multilevel preconditioned solvers, SIAM J. Sci. Comput., 23 (2001), 1053–1076.Google Scholar
[26]Szabó, B. and Babžka, I., Finite Element Analysis, John Wiley & Sons, New York, 1991.Google Scholar
[27]Webb, J. P., Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements, IEEE Trans. Antenna. Propaga., 47 (1999), 1244–1253.Google Scholar
[28]Whitney, H., Geometric Integration Theory, Princeton University Press, Princeton, 1957.Google Scholar
[29]Xin, J., Pinchedez, K. and Flaherty, J. E., Implementation of hierarchical bases in FEMLAB for simplicial elements, ACM Trans. Math. Software., 31 (2005), 187–200.Google Scholar