The discrete compactness property for anisotropic edge elements on polyhedral domains∗

Ariel Luis Lombardi

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

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

Abstract

top
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

top

Lombardi, Ariel Luis. "The discrete compactness property for anisotropic edge elements on polyhedral domains∗." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2012): 169-181. <http://eudml.org/doc/222148>.

@article{Lombardi2012,
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},
keywords = {Discrete compactness property; edge elements; anisotropic finite elements; Maxwell equations; discrete compactness property},
language = {eng},
month = {8},
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/222148},
volume = {47},
year = {2012},
}

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
DA - 2012/8//
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; discrete compactness property
UR - http://eudml.org/doc/222148
ER -

References

top
  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.  
  2. D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math.87 (2000) 229–246.  
  3. D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer.19 (2010) 1–120.  
  4. A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math.101 (2005) 29–65.  
  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.  
  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.  
  7. V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Applications. Springer-Verlag, Berlin (1986).  
  8. R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer.11 (2002) 237–339.  
  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.  
  10. M. Krízek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal.29 (1992) 513–520.  
  11. R. Leis, Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York (1986).  
  12. A.L. Lombardi, Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal.31 (2011) 1683–1712.  
  13. P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003).  
  14. P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in R3. Math. Comp.70 (2001) 507–523.  
  15. J.C. Nédélec, Mixed finite elements in R3. Numer. Math.35 (1980) 315–341.  
  16. S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal.39 (2001) 784–816.  
  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).  
  18. Ch. Weber, A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci.2 (1980) 12–25.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.