Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T05:52:00.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Some mixed finite element methodson anisotropic meshes

Published online by Cambridge University Press:  15 April 2002

Mohamed Farhloul ,
Serge Nicaise  and
Luc Paquet
Mohamed Farhloul
Affiliation:
Université de Moncton, Département de Mathématiques et de Statistique, N.B., E1A 3 E9, Moncton, Canada. (farhlom@umoncton.ca)
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, 59313 Valenciennes Cedex 9, France. (snicaise@univ-valenciennes.fr),
Luc Paquet
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, 59313 Valenciennes Cedex 9, France. (Luc.Paquet@univ-valenciennes.fr)
Get access

Abstract

The paper deals with some mixed finite element methods on a classof anisotropic meshes based on tetrahedra and prismatic (pentahedral)elements. Anisotropic localinterpolation error estimates are derived in some anisotropic weighted Sobolevspaces. As particularapplications, the numerical approximation by mixed methods of the Laplace equationin domainswith edges is investigated where anisotropic finiteelement meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of thesolutions.

Keywords

Anisotropic meshRaviart-Thomas elementanisotropic interpolation error estimateLaplace equationedge singularitymixed FEM.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 5 , September 2001 , pp. 907 - 920
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acosta, G. and Durán, R.G., The maximum angle condition for mixed and non-conforming elements, application to the Stokes equations. SIAM J. Numer. Anal. 37 (1999) 18-36. CrossRef
Th. Apel, Anisotropic Finite Elements: Local Estimates and Applications, in Advances in Numerical Mathematics, Teubner, Eds., Stuttgart (1999).
Th. Apel, M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing 47 (1992) 277-293. CrossRef
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.
Th. Apel, 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. 3.0.CO;2-R>CrossRef
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.
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. 3.0.CO;2-S>CrossRef
Babuska, I. and Aziz, A.K., On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976) 214-226. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, in Springer Series in Computational Mathematics 15, Springer-Verlag, Berlin (1991).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
M. Dauge, Elliptic Boundary Value Problems on Corner Domains, in Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin, Heidelberg, New York (1988).
Dauge, M., Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners I. SIAM J. Math. Anal. 20 (1989) 74-97. CrossRef
Farhloul, M., Mixed and nonconforming finite element methods for the Stokes problem. Can. Appl. Math. Quart. 3 (1995) 399-418.
Farhloul, M. and Fortin, M., A new mixed finite element for the Stokes and elasticity problems. SIAM J. Numer. Anal. 30 (1993) 971-990. CrossRef
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).
P. Grisvard, Elliptic Problems in NonSmooth Domains, in Monographs and Studies in Mathematics 24, Pitman, Boston (1985).
Lubuma, J. and Nicaise, S., Dirichlet problems in polyhedral domains II: Approximation by FEM and BEM. J. Comp. Appl. Math. 61 (1995) 13-27. CrossRef
Nédélec, J.-C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315-341. CrossRef
Nédélec, J.-C., A new family of mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 50 (1986) 57-81. CrossRef
S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. Submitted to SIAM J. Numer. Anal.
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.
Raugel, G., Résolution numérique par une méthode d'éléments finis du problème Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A 286 (1978) 791-794.
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.
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.
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.
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.
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).