zbMATH — the first resource for mathematics

Méthodes d’éléments finis quasilinéaires en déplacement pour létude de milieux incompressibles. (French) Zbl 0538.76037
This is a theoretical paper treating Stokes’ problem for fluid dynamics or linear incompressible elasticity where the incompressibility condition plays an important role. Simplicial finite elements with a non-symmetric structure with respect to the pressure space are introduced. The corresponding convergence analysis demonstrates the excellent properties of these elements. Numerical results will be presented in a forthcoming paper.
Reviewer: W.Schönauer

76D07 Stokes and related (Oseen, etc.) flows
74S05 Finite element methods applied to problems in solid mechanics
76M99 Basic methods in fluid mechanics
Full Text: DOI EuDML
[1] R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975. Zbl0314.46030 MR450957 · Zbl 0314.46030
[2] A. K. Aziz and I. Babuska, Survey lectures on the mathematical foundations of the finite element method, in: The Mathematical Foundations of the Finite Element Method with Applications to Partial Biffer ential Equations, edited by by A. K. Aziz, Academic Press, New York, 1972, pp. 3-359. Zbl0268.65052 MR421106 · Zbl 0268.65052
[3] [3] F. BREZZI, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Analyse Numérique 8-R2, 1974, pp. 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047 · eudml:193255
[4] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[5] J. F. DEBONGNIE, Sur la formulation de Herrmann pour l’étude des solides incompressibles, Journal de Mécanique, Vol. 17, n^\circ 4, 1978, pp. 531-557. Zbl0413.73015 · Zbl 0413.73015
[6] M. ORTIN, Calcul Numérique des Écoulements de Fluides de Bingham et des Fluides Incompressibles par la méthode des Éléments Finis, Thèse de Doctorat d’État ès Sciences, Université Paris VI, 1972.
[7] V. GIRAULT and P. A. AVIART, Finite Element Approximation of the Navier-Stokes Equations, Lecture notes in Mathematics, Springer Verlag, Berlin, 1979. Zbl0413.65081 MR548867 · Zbl 0413.65081 · doi:10.1007/BFb0063447
[8] R. GLOWINSKI, J. L. LIONS and R. RÉMOLIÈRES, Analyse Numérique des Inéquations Variationnelles, Éditions Dunod-Bordas, Paris, 1976. Zbl0358.65091 · Zbl 0358.65091
[9] P. GRISVARD, Behavior of the solutions of an elliptic boundary value problem in a polygonal or a polyhedral domain, in : Numerical Solution of Partial Differ ential Equations, III SYNSPADE 1975, edited by B. Hubbard, Academic Press, NewYork, 1976. Zbl0361.35022 MR466912 · Zbl 0361.35022
[10] C. JOHNSON and . PITKÂRANTA, Analysis of some mixed finite element methods related to reduced integration, Research Report 80.02 R of the Department of Computer Sciences of the Chalmers University of Technology and the University of Göteborg, 1980. Zbl0482.65058 · Zbl 0482.65058 · doi:10.2307/2007276
[11] . A. LADYZHENSKAYA and N. N. URAL’CEVA, Équations aux Dérivées Partielles de type Elliptique, Dunod, Paris, 1968. Zbl0164.13001 MR239273 · Zbl 0164.13001
[12] T. ODEN, Finite Eléments of Nonlinear Continua, McGraw Hill, New York, 1972. Zbl0235.73038 · Zbl 0235.73038
[13] J. E. OSBORH, Regularity of Solutions of the Stokes problem in a polygonal domain, in : Numerical Solution of Partial Differ ential Equations, III Synspade 1975, edited by B. Hubbard, Academic Press, New York, 1976. Zbl0344.65049 MR467032 · Zbl 0344.65049
[14] J. PITKARÄNTA, On a mixed finite element method for the Stokes problem in R 3 , RAIRO - Analyse Numérique (à paraître). Zbl0488.76039 · Zbl 0488.76039 · eudml:193400
[15] G. RAUGEL, Résolution Numérique de Problèmes Elliptiques dans des domaines avec Coins, Thèse de Doctorat de Troisième Cycle, Université de Rennes, 1978.
[16] V. RUAS, A class of asymmetrie finite element methods for solving finite incompressible elasticity problems, Comp. Meths. in Appl. Mechs. and Engin., 27, 1981,pp. 319-343. Zbl0467.73098 MR632279 · Zbl 0467.73098 · doi:10.1016/0045-7825(81)90136-5
[17] V. RUAS, Une méthode d’éléments finis non conformes en vitesse pour le problème de Stokes tridimensionnel, Matematica Aplicada e Computacional, V. 1, 1, pp. 53.74, 1982. Zbl0489.76049 MR667618 · Zbl 0489.76049
[18] V. RUAS, Méthodes d’Éléments Finis en Élasticité Incompressible Non Linéaire et Diverses Contributions à l’Approximation des Problèmes aux Limites, Thèse de Doctorat d’État ès Sciences, Université Pierre et Marie Curie, Paris, janvier 1982.
[19] G. STRANG and J. Fix, An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, N.J., 1973. Zbl0356.65096 MR443377 · Zbl 0356.65096
[20] R. TEMAM, Navier-Stokes Equations, North Holland, Amsterdam, 1977. Zbl0383.35057 MR603444 · Zbl 0383.35057
[21] J. M. THOMAS, Sur l’Analyse Numérique de Méthodes d’Éléments Finis Hybrides et Mixtes, Thèse de Doctorat d’État ès Sciences, Université Pierre et Marie Curie, Paris, 1977.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.