Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A dual finite element complex on the barycentric refinement

Authors: Annalisa Buffa and Snorre H. Christiansen
Journal: Math. Comp. 76 (2007), 1743-1769
MSC (2000): Primary 65N30, 65N38
Published electronically: May 3, 2007
MathSciNet review: 2336266
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex $ X^\bullet$ centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex $ Y^\bullet$ of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the $ \mathrm{L}^2$ duality is non-degenerate on $ Y^i \times X^{2-i}$ for each $ i\in \{0,1,2\}$. In particular $ Y^1$ is a space of $ \mathrm{curl}$-conforming vector fields which is $ \mathrm{L}^2$ dual to Raviart-Thomas $ \operatorname{div}$-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

References [Enhancements On Off] (What's this?)

  • 1. A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comput. 68 (1999), No.226, 607-631. MR 1609607 (99i:78002)
  • 2. C. Amrouche, C. Bernardi, M Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci. 21 (1998), 823-864. MR 1626990 (99e:35037)
  • 3. Douglas N. Arnold, Differential complexes and numerical stability, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 137-157. MR 1989182 (2004h:65115)
  • 4. A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic maxwell equations by a boundary finite element method. I: The continuous problem. II: The discrete problem, Math. Comp. 43 (1984), no. 167, 29-46 and 47-68. MR 0744923 (86i:65071a); MR 0744924 (86i:65071b)
  • 5. A. Bossavit, Mixed finite elements and the complex of Whitney forms, The mathematics of finite elements and applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 137-144. MR 956893 (89k:58028)
  • 6. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, second ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
  • 7. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, vol. 15, Springer-Verlag, Berlin, 1991. MR 1115205 (92d:65187)
  • 8. A. Buffa, Traces theorems for functional spaces related to Maxwell equations: an overview, Computational Electromagnetics (Kiel, Germany) (C. Castersen et al., ed.), Lectures notes in Computational Science and Engineering, vol. 28, Springer-Verlag, 2001.
  • 9. A. Buffa and S. H. Christiansen, The electric field integral equation on Lipschitz screens: definitions and numerical approximation, Numer. Math. 94 (2003), no. 2, 229-267. MR 1974555 (2004c:78016)
  • 10. -, A dual finite element complex on the barycentric refinement, Comptes Rendus Mathematique 340 (2005), no. 6, 461-464. MR 2135331
  • 11. A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations. Part I: An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appl. Sci. 21 (2001), no. 1, 9-30. MR 1809491 (2002b:78024)
  • 12. -, On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci. 21 (2001), no. 1, 31-48. MR 1809492 (2002b:78025)
  • 13. A. Buffa, R. Hiptmair, T. von Petersdorff, and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math. 95 (2003), no. 3, 459-485. MR 2012928 (2004i:65131)
  • 14. Snorre H. Christiansen and Jean-Claude Nédélec, A preconditioner for the electric field integral equation based on Calderon formulas, SIAM J. Numer. Anal. 40 (2002), no. 3, 1100-1135 (electronic). MR 1949407 (2003m:65235)
  • 15. Snorre Harald Christiansen, Résolution des équations intégrales pour la diffraction d'ondes acoustiques et électromagnétiques, Ph.D. thesis, Ecole Polytechnique, Palaiseau, France, 2001.
  • 16. Snorre Harald Christiansen and Jean-Claude Nédélec, Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l'acoustique, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 617-622. MR 1760450 (2001c:76088)
  • 17. Ph. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • 18. R. Coifman, V. Rokhlin, and S. Wandzura, The fast multipole method for the wave equation: a pedestrian prescription, IEEE Trans. Ant. Prop. 35 (1993), 7-12.
  • 19. M. Crouzeix and V. Thomée, The stability in $ L\sb p$ and $ W\sp 1\sb p$ of the $ L\sb 2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521-532. MR 878688 (88f:41016)
  • 20. Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475 (2003m:18001)
  • 21. R. Hiptmair and C. Schwab, Natural boundary element methods for the electric field integral equation on polyhedra, SIAM J. Numer. Anal. 40 (2002), no. 1, 66-86 (electronic). MR 1921910 (2003j:78059)
  • 22. Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183 (87e:15001)
  • 23. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
  • 24. P Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003. MR 2059447 (2005d:65003)
  • 25. V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 82-93. MR 1256528 (95d:35034)
  • 26. J. Song, C. Lu, and W.C. Chew, Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects, IEEE Trans. Ant. Prop. 45 (1997), 1488-1493.
  • 27. O. Steinbach and W. L. Wendland, The construction of some efficient preconditioners in the boundary element method, Adv. Comput. Math. 9 (1998), no. 1-2, 191-216. MR 1662766 (99j:65219)
  • 28. Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996, Basic theory. MR 1395147 (98b:35002a)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N38

Retrieve articles in all journals with MSC (2000): 65N30, 65N38

Additional Information

Annalisa Buffa
Affiliation: Istituto di Matematica Applicata e Tecnologie Informatiche - CNR, Via Ferrata 1, 27100 Pavia, Italy

Snorre H. Christiansen
Affiliation: CMA c/o Matematisk Institutt, PB 1053 Blindern, Universitetet i Oslo, NO-0316 Oslo, Norway

Received by editor(s): July 6, 2005
Received by editor(s) in revised form: July 25, 2006
Published electronically: May 3, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society