A dual finite element complex on the barycentric refinement
HTML articles powered by AMS MathViewer
- by Annalisa Buffa and Snorre H. Christiansen PDF
- Math. Comp. 76 (2007), 1743-1769 Request permission
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
- Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607, DOI 10.1090/S0025-5718-99-01013-3
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
- Douglas N. Arnold, Differential complexes and numerical stability, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 137–157. MR 1989182
- 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, Math. Comp. 43 (1984), no. 167, 29–46. MR 744923, DOI 10.1090/S0025-5718-1984-0744923-1
- 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
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- 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.
- 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, DOI 10.1007/s00211-002-0422-0
- Annalisa Buffa and Snorre H. Christiansen, A dual finite element complex on the barycentric refinement, C. R. Math. Acad. Sci. Paris 340 (2005), no. 6, 461–464 (English, with English and French summaries). MR 2135331, DOI 10.1016/j.crma.2004.12.022
- A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci. 24 (2001), no. 1, 9–30. MR 1809491, DOI 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
- A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci. 24 (2001), no. 1, 31–48. MR 1809492, DOI 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
- 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, DOI 10.1007/s00211-002-0407-z
- 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. MR 1949407, DOI 10.1137/S0036142901388731
- Snorre Harald Christiansen, Résolution des équations intégrales pour la diffraction d’ondes acoustiques et électromagnétiques, Ph.D. thesis, Ecole Polytechnique, Palaiseau, France, 2001.
- 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 (French, with English and French summaries). MR 1760450, DOI 10.1016/S0764-4442(00)00225-1
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- 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.
- M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
- Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475, DOI 10.1007/978-3-662-12492-5
- 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. MR 1921910, DOI 10.1137/S0036142901387580
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
- Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
- 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, DOI 10.1006/acha.1993.1006
- 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.
- 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. Numerical treatment of boundary integral equations. MR 1662766, DOI 10.1023/A:1018937506719
- Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR 1395147, DOI 10.1007/978-1-4684-9320-7
Additional Information
- Annalisa Buffa
- Affiliation: Istituto di Matematica Applicata e Tecnologie Informatiche - CNR, Via Ferrata 1, 27100 Pavia, Italy
- Email: annalisa@imati.cnr.it
- Snorre H. Christiansen
- Affiliation: CMA c/o Matematisk Institutt, PB 1053 Blindern, Universitetet i Oslo, NO-0316 Oslo, Norway
- MR Author ID: 663397
- Email: snorrec@math.uio.no
- Received by editor(s): July 6, 2005
- Received by editor(s) in revised form: July 25, 2006
- Published electronically: May 3, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1743-1769
- MSC (2000): Primary 65N30, 65N38
- DOI: https://doi.org/10.1090/S0025-5718-07-01965-5
- MathSciNet review: 2336266