Commuting diagrams for the TNT elements on cubes
Authors:
Bernardo Cockburn and Weifeng Qiu
Journal:
Math. Comp. 83 (2014), 603-633
MSC (2010):
Primary 65N30, 65L12
DOI:
https://doi.org/10.1090/S0025-5718-2013-02729-9
Published electronically:
June 18, 2013
MathSciNet review:
3143686
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We present commuting diagrams for the de Rham complex for new elements defined on cubes which use tensor product spaces. The distinctive feature of these elements is that, in sharp contrast with previously known results, they have the TiNiest spaces containing Tensor product spaces of polynomials of degree $k$, hence their acronym TNT. In fact, the local spaces of the TNT elements differ from the standard tensor product spaces by spaces whose dimension is a small number independent of the degree $k$. Such a number is 7 (the number of vertices of the cube minus one) for the space associated with the divergence operator, 18 (the number of faces of the cube times the number of vertices of a face minus one) for the space associated with the curl operator, and 12 (the number of edges of the cube times the number of vertices of an edge minus one) for the space associated with the gradient operator.
- Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749β1779. MR 1885715, DOI https://doi.org/10.1137/S0036142901384162
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1β155. MR 2269741, DOI https://doi.org/10.1017/S0962492906210018
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281β354. MR 2594630, DOI https://doi.org/10.1090/S0273-0979-10-01278-4
- Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO ModΓ©l. Math. Anal. NumΓ©r. 21 (1987), no. 4, 581β604 (English, with French summary). MR 921828, DOI https://doi.org/10.1051/m2an/1987210405811
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217β235. MR 799685, DOI https://doi.org/10.1007/BF01389710
- C. D. Cantwell, S. J. Sherwin, R. M. Kirby, and P. H. J. Kelly, From $h$ to $p$ efficiently: strategy selection for operator evaluation on hexahedral and tetrahedral elements, Comput. & Fluids 43 (2011), 23β28. MR 2775064, DOI https://doi.org/10.1016/j.compfluid.2010.08.012
- Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327β1353. MR 2904581, DOI https://doi.org/10.1090/S0025-5718-2011-02550-0
- ---, Superconvergent HDG methods on isoparametric elements for second-order elliptic problems, SIAM J. Numer. Anal. To appear.
- Richard S. Falk, Paolo Gatto, and Peter Monk, Hexahedral $H(\rm div)$ and $H(\rm curl)$ finite elements, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 1, 115β143. MR 2781133, DOI https://doi.org/10.1051/m2an/2010034
- J.-C. NΓ©dΓ©lec, Mixed finite elements in ${\bf R}^{3}$, Numer. Math. 35 (1980), no. 3, 315β341. MR 592160, DOI https://doi.org/10.1007/BF01396415
- Weifeng Qiu and Leszek Demkowicz, Mixed $hp$-finite element method for linear elasticity with weakly imposed symmetry, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 47-48, 3682β3701. MR 2557491, DOI https://doi.org/10.1016/j.cma.2009.07.010
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292β315. Lecture Notes in Math., Vol. 606. MR 0483555
- Peter E. J. Vos, Spencer J. Sherwin, and Robert M. Kirby, From $h$ to $p$ efficiently: implementing finite and spectral/$hp$ element methods to achieve optimal performance for low- and high-order discretisations, J. Comput. Phys. 229 (2010), no. 13, 5161β5181. MR 2643647, DOI https://doi.org/10.1016/j.jcp.2010.03.031
- T. Warburton, L. F. Pavarino, and J. S. Hesthaven, A pseudo-spectral scheme for the incompressible Navier-Stokes equations using unstructured nodal elements, J. Comput. Phys. 164 (2000), no. 1, 1β21. MR 1786240, DOI https://doi.org/10.1006/jcph.2000.6587
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Additional Information
Bernardo Cockburn
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
cockburn@math.umn.edu
Weifeng Qiu
Affiliation:
Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID:
845089
Email:
qiuxa001@ima.umn.edu, qiuw78@gmail.com
Keywords:
Commuting diagrams,
cubic element,
tensor product spaces
Received by editor(s):
July 12, 2011
Received by editor(s) in revised form:
March 5, 2012, March 30, 2012, and June 21, 2012
Published electronically:
June 18, 2013
Additional Notes:
The first author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute.
The second author gratefully acknowledges the collaboration opportunities provided by IMA (Minneapolis) during their 2010β2012 program. \indent Corresponding author: Weifeng Qiu
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.