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

  • Jianguo Xin (a1) and Wei Cai (a1)


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].


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[1]Abdul-Rahman, R. and Kasper, M., Orthogonal hierarchical Nédélec elements, IEEE Trans. Magn., 44 (2008), 12101213.
[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), 29252941.
[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), 67096733.
[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.
[5]Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numer. Methods. Engrg., 58 (2003), 21032130.
[6]Babžka, I., Szabo, B. A. and Katz, I. N., The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515545.
[7]Bittencourt, M. L., Fully tensorial nodal and modal shape functions for triangles and tetra-hedra, Int. J. Numer. Methods. Engrg., 63 (2005), 15301558.
[8]Bossavit, A., Computational Electromagnetism, Academic Press, New York, 1998.
[9]Dunkl, C. F. and Xu, Y., Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications, 81, Cambridge University Press, Cambridge, 2001.
[10]Zhang, F. (Editor), The Schur Complement and Its Applications, Springer-Verlag, New York, 2005.
[11]Gopalakrishnan, J., García-Castillo, L. E. and Demkowicz, L. F., Nédélec spaces in affine coordinates, Comput. Math. Appl., 49 (2005), 12851294.
[12]Hiptmair, R., Canonical construction of finite elements, Math. Comput., 68 (1999), 13251346.
[13]Hiptmair, R., Higher order Whitney forms, PIER., 32 (2001), 271299.
[14]Ingelström, P., A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes, IEEE Trans. Microw. Theor. Tech., 54 (2006), 106114.
[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), 29852995.
[16]Karniadakis, G. E. and Sherwin, S. J., Spectral/hp Element Methods for CFD, Oxford University Press, New York, 1999.
[17]Nédélec, J. C., Mixed finite elements in R3, Numer. Math., 35 (1980), 315341.
[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), 147180.
[19]Rapetti, F., High order edge elements on simplicial meshes, ESAIM: M2AN, 41 (2007), 1001–1020.
[20]Rapetti, F. and Bossavit, A., Whitney forms of higher degree, SIAM J. Numer. Anal., 47 (2009), 23692386.
[21]Ren, Z. and Ida, N., High order differential form-based elements for the computation of electromagnetic field, IEEE Trans. Magn., 36 (2000), 14721478.
[22]Schöberl, J. and Zaglmayr, S., High order Nédélec elements with local complete sequence properties, Compel., 24 (2005), 374384.
[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), 14891505.
[24]Shreshevskii, I. A., Orthogonalization of graded sets of vectors, J. Nonlinear. Math. Phys., 8 (2001), 5458.
[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), 10531076.
[26]Szabó, B. and Babžka, I., Finite Element Analysis, John Wiley & Sons, New York, 1991.
[27]Webb, J. P., Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements, IEEE Trans. Antenna. Propaga., 47 (1999), 12441253.
[28]Whitney, H., Geometric Integration Theory, Princeton University Press, Princeton, 1957.
[29]Xin, J., Pinchedez, K. and Flaherty, J. E., Implementation of hierarchical bases in FEMLAB for simplicial elements, ACM Trans. Math. Software., 31 (2005), 187200.


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

  • Jianguo Xin (a1) and Wei Cai (a1)


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