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Conforming Hierarchical Basis Functions

Published online by Cambridge University Press:  20 August 2015

M. J. Bluck*
Affiliation:
Department of Mechanical Engineering, Imperial College, London, SW7 2AZ, UK
*
*Corresponding author.Email:m.bluck@imperial.ac.uk
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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 and Arnold et al. 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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Abdul-Rahman, R. and Kasper, M.. Orthogonal hierarchical nedelec elements. IEEE Trans. Magn. (USA), 44(6): 12101213, 2008.Google Scholar
[2]Andersen, L.S. and Volakis, J.L.. Hierarchical tangential vector finite elements for tetrahedra. IEEE Microw. Guid. Wave Lett. (USA), 8(3): 127129, 1998.Google Scholar
[3]Andersen, L.S. and Volakis, J.L.. Development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics. IEEE Trans. Antennas Propag. (USA), 47(1): 112120, 1999.Google Scholar
[4]Arnold, D.N., Falk, R.S., and Winther, R.. Finite element exterior calculus, homological techniques, and applications. Acta Numer. (UK), 15: 1155, 2006.Google Scholar
[5]Bluck, M. J., Hatzipetros, A., and Walker, S. P.. Applications of differential forms to boundary integral equations. IEEE Transactions on Antennas and Propagation, 54(6): 17811796, 2006.CrossRefGoogle Scholar
[6]Bluck, M. J. and Walker, S. P.. Polynomial basis functions on pyramidal elements. Communications in Numerical Methods in Engineering, 2007.Google Scholar
[7]Bossavit, A.. Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc. A, Phys. Sci. Meas. Instrum. Manage. Educ. Rev. (UK), 135(8): 493500, 1988.Google Scholar
[8]Bossavit, A.. Simplicial finite elements for scattering problems in electromagnetism. Comput. Methods Appl. Mech. Eng. (Netherlands), 76(3): 299316, 1989.Google Scholar
[9]Bossavit, A.. Generating whitney forms of polynomial degree one and higher. IEEE Transactions on Magnetics, 38(2, pt.1), 2002.Google Scholar
[10]Eastwood, J.W. and Morgan, J.G.. Higher-order basis functions for mom calculations. IET Sci. Meas. Technol. (UK), 2(6): 379386, 2008.Google Scholar
[11]Falk, R. S., Gatto, P., and Monk, P.. Hexahedral h(div) and h(curl) finite elements. ESAIM: Mathematical Modelling and Numerical Analysis, 45(1): 115143, 2011.Google Scholar
[12]Gradinaru, V. and Hiptmair, R.. Whitney elements on pyramids. Electron. Trans. Numer. Anal., 8: 154168, 1999.Google Scholar
[13]Graglia, R. D. and Gheorma, I. L.. Higher order interpolatory vector bases on pyramidal elements. IEEE Transactions on Antennas and Propagation, 47(5): 775, 1999.Google Scholar
[14]Graglia, R. D., Wilton, D. R., and Peterson, A. F.. Higher order interpolatory vector bases for computational electromagnetics. IEEE Transactions on Antennas and Propagation, 45(3): 329, 1997.Google Scholar
[15]Graglia, R.D., Peterson, A.F., and Andriulli, F.P.. Curl-conforming hierarchical vector bases for triangles and tetrahedra. Antennas and Propagation, IEEE Transactions on, 59(3): 950959, March 2011.Google Scholar
[16]Hiptmair, R.. Canonical construction of finite elements. Mathematics of Computation, 68(228): 1325, 1999.Google Scholar
[17]Hiptmair, R.. Higher order whitney forms. Journal of Electromagnetic Waves and Applications, 15(3): 341, 2001.Google Scholar
[18]Hiptmair, R.. Finite elements in computational electromagnetism. Acta Numerica, 11: 237, 2002.Google Scholar
[19]Ilic, M.M. and Notaros, B.M.. Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling. IEEE Trans. Microw. Theory Tech. (USA), 51(3): 10261033, 2003.Google Scholar
[20]Ilic, M.M. and Notaros, B.M.. Higher order large-domain hierarchical fem technique for electromagnetic modeling using legendre basis functions on generalized hexahedra. Electromagnetics (UK), 26(7): 517529, 2006.Google Scholar
[21]Ingelstrom, P.. A new set of h(curl)-conforming hierarchical basis functions for tetrahedral meshes. IEEE Trans. Microw. Theory Tech. (USA), 54(1): 106114, 2006.Google Scholar
[22]Ingelstrom, P., Hill, V., and Dyczij-Edlinger, R.. Comparison of hierarchical basis functions for efficient multilevel solvers. IET Sci. Meas. Technol. (UK), 1(1): 4852, 2007.Google Scholar
[23]Nakahara, M.. Geometry, topology and physics. Second edition. IOP Publishing, 2003.Google Scholar
[24]Nedelec, J. C.. Mixed finite elements in r/sup 3. Numerische Mathematik, 35(3): 315, 1980.Google Scholar
[25]Schoberl, J. and Zaglmayr, S.. High order nedelec elements with local complete sequence property. COMPEL, 24(2): 374384, 2005.CrossRefGoogle Scholar
[26]Showalter, R.E.. Hilbert Space Methods for Partial Differential Equations. Pitman Publishing, London, 1977.Google Scholar
[27]Webb, J.P.. Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. Antennas Propag. (USA), 47(8): 12441253, 1999.Google Scholar
[28]Xin, J. and Cai, W.. A well-conditioned hierarchical basis for triangular h(curl)-conforming elements. Commun. Comput. Phys., 9(3): 780806, 2011.Google Scholar
[29]Xin, J., Guo, N., and Cai, W.. On the construction of well-conditioned hierarchical bases for tetrahedral h(curl)-conforming nedelec elements. Journal of Computational Mathematics, 29(5): 526542, 2011.Google Scholar