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

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 , 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 . 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.

**1.**D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini,*Unified analysis of discontinuous Galerkin methods for elliptic problems*, SIAM J. Numer. Anal.**39**(2002), 1749-1779. MR**1885715 (2002k:65183)****2.**D. N. Arnold, R. Falk, and R. Winther,*Finite element exterior calculus, homological techniques, and applications*, Acta Numer.**15**(2006), 1-155. MR**2269741 (2007j:58002)****3.**D. N. Arnold, R. Falk, and R. Winther,*Finite element exterior calculus: from Hodge theory to numerical stability*, Bull. Amer. Math. Soc.(N.S.)**47**(2010), 281-354. MR**2594630 (2011f:58005)****4.**F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini,*Efficient rectangular mixed finite element methods in two and three space variables*, RAIRO Modél. Math. Anal. Numér.**21**(1987), 581-604. MR**921828 (88j:65249)****5.**F. Brezzi, J. Douglas, Jr., and L. D. Marini,*Two families of mixed finite elements for second order elliptic problems*, Numer. Math.**47**(1985), 217-235. MR**799685 (87g:65133)****6.**C.D. Cantwell, S.J. Sherwin, R.M. Kirby, and P.H.J. Kelly,*From to efficiency: Strategy selection for operator evaluation on hexahedral and tetrahedral elements*, Computers and Fluids**43**(2011), 23-28. MR**2775064****7.**B. Cockburn, W. Qiu, and K. Shi,*Conditions for superconvergence of HDG methods for second-order elliptic problems*, Math. Comp.**81**(2012) no. 279, 1327-1353. MR**2904581****8.**-,*Superconvergent HDG methods on isoparametric elements for second-order elliptic problems*, SIAM J. Numer. Anal. To appear.**9.**R. Falk, P. Gatto, and P. Monk,*Hexahedral and finite elements*, MAN (2010). MR**2781133 (2012d:65275)****10.**J.-C. Nédélec,*Mixed finite elements in*, Numer. Math.**35**(1980), 315-341. MR**592160 (81k:65125)****11.**W. Qiu, and L. Demkowicz,*Mixed -finite element method for linear elasticity with weakly imposed symmetry*, Comput. Methods Appl. Mech. Engrg.**198**(2009), 3682-3701. MR**2557491 (2011a:74011)****12.**P. A. Raviart and J. M. Thomas,*A mixed finite element method for second order elliptic problems*, Mathematical Aspects of Finite Element Method, Lecture Notes in Math. 606 (I. Galligani and E. Magenes, eds.), Springer-Verlag, New York, 1977, pp. 292-315. MR**0483555 (58:3547)****13.**P.E.J. Vos, S.J. Sherwin, and R.M. Kirby,*From to efficiency: Implementing finite and spectral element methods to achieve optimal performance for low- and high-order discretizations*, J. Comput. Phys.**229**(2010), 5161-5181. MR**2643647 (2011i:65225)****14.**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), 1-21. MR**1786240 (2001f:76062)**

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
65N30,
65L12

Retrieve articles in all journals with MSC (2010): 65N30, 65L12

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

Email:
qiuxa001@ima.umn.edu, qiuw78@gmail.com

DOI:
https://doi.org/10.1090/S0025-5718-2013-02729-9

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 2010–2012 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.