Some mixed finite element methods on anisotropic meshes

Mohamed Farhloul; Serge Nicaise; Luc Paquet

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 5, page 907-920
  • ISSN: 0764-583X

Abstract

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The paper deals with some mixed finite element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived in some anisotropic weighted Sobolev spaces. As particular applications, the numerical approximation by mixed methods of the Laplace equation in domains with edges is investigated where anisotropic finite element meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of the solutions.

How to cite

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Farhloul, Mohamed, Nicaise, Serge, and Paquet, Luc. "Some mixed finite element methods on anisotropic meshes." ESAIM: Mathematical Modelling and Numerical Analysis 35.5 (2010): 907-920. <http://eudml.org/doc/197402>.

@article{Farhloul2010,
abstract = { The paper deals with some mixed finite element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived in some anisotropic weighted Sobolev spaces. As particular applications, the numerical approximation by mixed methods of the Laplace equation in domains with edges is investigated where anisotropic finite element meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of the solutions. },
author = {Farhloul, Mohamed, Nicaise, Serge, Paquet, Luc},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Anisotropic mesh; Raviart-Thomas element; anisotropic interpolation error estimate; Laplace equation; edge singularity; mixed FEM.; Laplace equation; mixed finite element methods; anisotropic meshes; error estimates},
language = {eng},
month = {3},
number = {5},
pages = {907-920},
publisher = {EDP Sciences},
title = {Some mixed finite element methods on anisotropic meshes},
url = {http://eudml.org/doc/197402},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Farhloul, Mohamed
AU - Nicaise, Serge
AU - Paquet, Luc
TI - Some mixed finite element methods on anisotropic meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 5
SP - 907
EP - 920
AB - The paper deals with some mixed finite element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived in some anisotropic weighted Sobolev spaces. As particular applications, the numerical approximation by mixed methods of the Laplace equation in domains with edges is investigated where anisotropic finite element meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of the solutions.
LA - eng
KW - Anisotropic mesh; Raviart-Thomas element; anisotropic interpolation error estimate; Laplace equation; edge singularity; mixed FEM.; Laplace equation; mixed finite element methods; anisotropic meshes; error estimates
UR - http://eudml.org/doc/197402
ER -

References

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  1. G. Acosta and R.G. Durán, The maximum angle condition for mixed and non-conforming elements, application to the Stokes equations. SIAM J. Numer. Anal.37 (1999) 18-36.  Zbl0948.65115
  2. Th. Apel, Anisotropic Finite Elements: Local Estimates and Applications, in Advances in Numerical Mathematics, Teubner, Eds., Stuttgart (1999).  
  3. Th. Apel and M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing47 (1992) 277-293.  Zbl0746.65077
  4. Th. Apel and S. Nicaise, Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes, in Partial Differential Equations and Functional Analysis (in memory of Pierre Grisvard), J. Céa, D. Chenais, G. Geymonat, and J.-L. Lions, Eds., Birkhäuser, Boston (1996) 18-34.  Zbl0854.35005
  5. Th. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci.21 (1998) 519-549.  Zbl0911.65107
  6. Th. Apel, S. Nicaise and J. Schöberl, A non-conforming FEM with anisotropic mesh grading for the Stokes problem in domains with edges. To appear in IMAJ Numer. Anal. Zbl0998.65116
  7. Th. Apel, A.-M. Sändig, J.R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci.19 (1996) 63-85.  Zbl0838.65109
  8. I. Babuska and A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal.13 (1976) 214-226.  Zbl0324.65046
  9. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, in Springer Series in Computational Mathematics 15, Springer-Verlag, Berlin (1991).  Zbl0788.73002
  10. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).  Zbl0383.65058
  11. M. Dauge, Elliptic Boundary Value Problems on Corner Domains, in Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin, Heidelberg, New York (1988).  Zbl0668.35001
  12. M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners I. SIAM J. Math. Anal.20 (1989) 74-97.  Zbl0681.35071
  13. M. Farhloul, Mixed and nonconforming finite element methods for the Stokes problem. Can. Appl. Math. Quart.3 (1995) 399-418.  Zbl0846.35100
  14. M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems. SIAM J. Numer. Anal.30 (1993) 971-990.  Zbl0777.76051
  15. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York (1986).  Zbl0413.65081
  16. P. Grisvard, Elliptic Problems in NonSmooth Domains, in Monographs and Studies in Mathematics 24, Pitman, Boston (1985).  Zbl0695.35060
  17. J. Lubuma and S. Nicaise, Dirichlet problems in polyhedral domains II: Approximation by FEM and BEM. J. Comp. Appl. Math.61 (1995) 13-27.  Zbl0840.65110
  18. J.-C. Nédélec, Mixed finite elements in 3 . Numer. Math.35 (1980) 315-341.  Zbl0419.65069
  19. J.-C. Nédélec, A new family of mixed finite elements in 3 . Numer. Math.50 (1986) 57-81.  Zbl0625.65107
  20. S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. Submitted to SIAM J. Numer. Anal. Zbl1001.65122
  21. L.A. Oganesyan and L.A. Rukhovets, Variational-difference methods for the solution of elliptic equations. Izd. Akad. Nauk Armyanskoi SSR, Jerevan (1979). In Russian.  Zbl0496.65053
  22. G. Raugel, Résolution numérique par une méthode d'éléments finis du problème Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A286 (1978) 791-794.  Zbl0377.65058
  23. P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, in Lecture Notes in Mathematics 606, I. Galligani and E. Magenes, Eds., Springer-Verlag, Berlin (1977) 292-315.  
  24. J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part 1), J.-L. Lions and P.G. Ciarlet, Eds., North-Holland Publishing Company, Amsterdam, New York, Oxford (1991) 523-639.  Zbl0875.65090
  25. H. El Sossa and L. Paquet, Refined mixed finite element method of the Dirichlet problem for the Laplace equation in a polygonal domain, in Rapport de Recherche 00-5, Laboratoire MACS, Université de Valenciennes, France. Submitted to Adv. Math. Sc. Appl. 
  26. H. El Sossa and L. Paquet, Méthodes d'éléments finis mixtes raffinés pour le problème de Stokes, in Rapport de Recherche 01-1, Laboratoire MACS, Université de Valenciennes, France.  
  27. J.M. Thomas, Sur l'analyse numérique des méthodes d'élements finis mixtes et hybrides. Thèse d'État, Université Pierre et Marie Curie, Paris (1977).  

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