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High Order Mass-Lumping Finite Elements on Simplexes

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

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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