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High Order Mass-Lumping Finite Elements on Simplexes

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical approximation and computational geometry

Published online by Cambridge University Press:  09 May 2017

Tao Cui ,
Wei Leng ,
Deng Lin ,
Shichao Ma  and
Linbo Zhang
Tao Cui*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, Beijing 100190, China
Wei Leng*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, Beijing 100190, China
Deng Lin*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Shichao Ma*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Linbo Zhang*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
*
*Corresponding author. Email addresses:tcui@lsec.cc.ac.cn (T. Cui), wleng@lsec.cc.ac.cn (W. Leng), lind@lsec.cc.ac.cn (D. Lin), masc@lsec.cc.ac.cn (S. C. Ma), zlb@lsec.cc.ac.cn (L. B. Zhang)
*Corresponding author. Email addresses:tcui@lsec.cc.ac.cn (T. Cui), wleng@lsec.cc.ac.cn (W. Leng), lind@lsec.cc.ac.cn (D. Lin), masc@lsec.cc.ac.cn (S. C. Ma), zlb@lsec.cc.ac.cn (L. B. Zhang)
*Corresponding author. Email addresses:tcui@lsec.cc.ac.cn (T. Cui), wleng@lsec.cc.ac.cn (W. Leng), lind@lsec.cc.ac.cn (D. Lin), masc@lsec.cc.ac.cn (S. C. Ma), zlb@lsec.cc.ac.cn (L. B. Zhang)
*Corresponding author. Email addresses:tcui@lsec.cc.ac.cn (T. Cui), wleng@lsec.cc.ac.cn (W. Leng), lind@lsec.cc.ac.cn (D. Lin), masc@lsec.cc.ac.cn (S. C. Ma), zlb@lsec.cc.ac.cn (L. B. Zhang)
*Corresponding author. Email addresses:tcui@lsec.cc.ac.cn (T. Cui), wleng@lsec.cc.ac.cn (W. Leng), lind@lsec.cc.ac.cn (D. Lin), masc@lsec.cc.ac.cn (S. C. Ma), zlb@lsec.cc.ac.cn (L. B. Zhang)
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Abstract

This paper is concerned with the construction of high order mass-lumping finite elements on simplexes and a program for computing mass-lumping finite elements on triangles and tetrahedra. The polynomial spaces for mass-lumping finite elements, as proposed in the literature, are presented and discussed. In particular, the unisolvence problem of symmetric point-sets for the polynomial spaces used in mass-lumping elements is addressed, and an interesting property of the unisolvent symmetric point-sets is observed and discussed. Though its theoretical proof is still lacking, this property seems to be true in general, and it can greatly reduce the number of cases to consider in the computations of mass-lumping elements. A program for computing mass-lumping finite elements on triangles and tetrahedra, derived from the code for computing numerical quadrature rules presented in [7], is introduced. New mass-lumping finite elements on triangles found using this program with higher orders, namely 7, 8 and 9, than those available in the literature are reported.

Keywords

Finite elementmass-lumpingsimplexnumerical quadraturemultivariate polynomial interpolationunisolvence

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65D05: Interpolation 65D32: Quadrature and cubature formulas
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 331 - 350
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Cohen, G., Joly, P., Roberts, J. E. and Tordjman, N., Higher order triangular finite elements with mass lumping for the wave equation, SIAM J. Numer. Anal., 38:6 (2001), pp. 2047–2078.CrossRefGoogle Scholar
[2] Chin-Joe-Kong, M. J. S., Mulder, W. A. and Van Veldhuizen, M., Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation, J. Eng. Math., 35 (1999), pp. 405–426.CrossRefGoogle Scholar
[3] Mulder, W. A., Higher-order mass-lumped finite elements for the wave equation, J. Comput. Acoustics, 9:2 (2001), pp. 671–680.CrossRefGoogle Scholar
[4] Jund, Sébastien, Méthodes D'éléments Finis D'ordre Élevé pour la Simulation Numérique de la Propagation D'ondes, PhD Thesis, Université Louis Pasteur–Strasbourg I, 2007.Google Scholar
[5] Mulder, W. A., New triangular mass-lumped finite elements of degree six for wave propagation, Prog. Electromagnetics Res., 141 (2013), pp. 671–692.CrossRefGoogle Scholar
[6] Giraldo, F. X. and Taylor, M. A., A diagonal-mass-matrix triangular-spectral-element method based on cubature points, J. Eng. Math., 56 (2006), pp. 307–322.CrossRefGoogle Scholar
[7] Zhang, L. B., Cui, T. and Liu, H., A set of symmetric quadrature rules on triangles and tetrahedra, J. Comput. Math., 27:1 (2009), pp. 89–96.Google Scholar
[8] Zhang, L. B., A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection, Numer. Math. Theory Methods Appl., 2 (2009), pp. 65–89.Google Scholar
[9] Zhang, L. B., Zheng, W. Y., Lu, B. Z., Cui, T., Leng, W. and Lin, D., The toolbox PHG and its applications, Science China Information Sciences (in Chinese), 46(10) (2016), pp. 1442–1464.Google Scholar
[10] Nicolaides, R. A., On a class of finite elements generated by Lagrange interpolation, SIAM J. Numer. Anal., 9 (1972), pp. 435–445.CrossRefGoogle Scholar
[11] Chung, K. C. and Yao, T. H., On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal., 14(4) (1977), pp. 735–743.CrossRefGoogle Scholar
[12] Roth, M. J., Nodal Configurations and Voronoi Tessellations for Triangular Spectral Elements, PhD Thesis, School of Earth and Ocean Sciences, University of Victoria, 2005.Google Scholar
[13] Taylor, M. A., Wingate, B. A. and Bos, L. P., A cardinal function algorithm for computing multivariate quadrature points, SIAM J. Numer. Anal., 45:1 (2007), pp. 193–205.CrossRefGoogle Scholar
[14] Papanicolopulos, S. A., Efficient computation of cubature rules with application to new asymmetric rules on the triangle, J. Comput. Appl. Math., 304 (2016), pp. 73–83.CrossRefGoogle Scholar