The discrete compactness property for anisotropic edge elements on polyhedral domains

Ariel Luis Lombardi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 169-181
  • ISSN: 0764-583X

Abstract

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We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.

How to cite

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Lombardi, Ariel Luis. "The discrete compactness property for anisotropic edge elements on polyhedral domains." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 169-181. <http://eudml.org/doc/273139>.

@article{Lombardi2013,
abstract = {We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.},
author = {Lombardi, Ariel Luis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {discrete compactness property; edge elements; anisotropic finite elements; Maxwell equations},
language = {eng},
number = {1},
pages = {169-181},
publisher = {EDP-Sciences},
title = {The discrete compactness property for anisotropic edge elements on polyhedral domains},
url = {http://eudml.org/doc/273139},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Lombardi, Ariel Luis
TI - The discrete compactness property for anisotropic edge elements on polyhedral domains
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 169
EP - 181
AB - We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.
LA - eng
KW - discrete compactness property; edge elements; anisotropic finite elements; Maxwell equations
UR - http://eudml.org/doc/273139
ER -

References

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  1. [1] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Meth. Appl. Sci.21 (1998) 519–549. Zbl0911.65107MR1615426
  2. [2] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math.87 (2000) 229–246. Zbl0967.65106MR1804657
  3. [3] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer.19 (2010) 1–120. Zbl1242.65110MR2652780
  4. [4] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math.101 (2005) 29–65. Zbl1116.78020MR2194717
  5. [5] S. Caorsi, P. Fernandes and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal.38 (2000) 580–607. Zbl1005.78012MR1770063
  6. [6] S. Caorsi, P. Fernandes and M. Raffetto, Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. Math. Model. Numer. Anal.35 (2001) 331–354. Zbl0993.78016MR1825702
  7. [7] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Applications. Springer-Verlag, Berlin (1986). Zbl0585.65077MR851383
  8. [8] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer.11 (2002) 237–339. Zbl1123.78320MR2009375
  9. [9] F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math.36 (1989) 479–490. Zbl0698.65067MR1039483
  10. [10] M. Krízek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal.29 (1992) 513–520. Zbl0755.41003MR1154279
  11. [11] R. Leis, Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York (1986). Zbl0599.35001MR841971
  12. [12] A.L. Lombardi, Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal.31 (2011) 1683–1712. Zbl1229.65196MR2846771
  13. [13] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003). Zbl1024.78009MR2059447
  14. [14] P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in R3. Math. Comp.70 (2001) 507–523. Zbl1035.65131MR1709155
  15. [15] J.C. Nédélec, Mixed finite elements in R3. Numer. Math.35 (1980) 315–341. Zbl0419.65069
  16. [16] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal.39 (2001) 784–816. Zbl1001.65122MR1860445
  17. [17] P. A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, edited by I. Galligani and E. Magenes. Lect. Notes Math. 606 (1977). Zbl0362.65089MR483555
  18. [18] Ch. Weber, A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci.2 (1980) 12–25. Zbl0432.35032MR561375

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