Summary
We begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brezzi, F., Fujii, H.: Mixed finite element approximations of the Von Kármán equations. Proceedings of the 4th L1BLICE Conference on Basic Problems of Numerical Analysis, Pilsen, Czechoslovakia, September 1978
Brezzi, F., Raviart, P-A: Mixed finite element methods for 4th order elliptic equations. Topics in Numerical Analysis III (J.J.H. Miller, ed.), pp. 33–56. London: Academic Press 1976
Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. R.A.I.R.O. Numer. Anal. (in press, 1980)
Fujii, H., Yamaguti, M.: Structure of singularities and its numerical realization in nonlinear elasticity, Research Report KSU/ICS 78-06, Kyoto Sangyo University
Girault, V., Raviart, P-A: Finite element approximation of the Navier-Stokes equations. Lecture Notes in Mathematics, No. 749. Heidelberg, New York: Springer 1979
Girault, V., Raviart, P-A: An analysis of a mixed finite element methods for the Navier-Stokes equations. Numer. Math.33, 235–271 (1979)
Girault, V., Raviart, P-A.: An analysis of an upwind scheme for the Navier-Stokes equations. (in press, 1980)
Grisvard, P.: Singularité des solutions du problème de Stokes dans un polygone. Publications de l'Université de Nice (1978)
Keller, H.B.: Approximation methods for nonlinear problems with applications to two-point boundary value problems. Math. Comput.29, 464–474 (1975)
Kondrat'ev, V.A.: Boundary value problems for elliptic equations in domains with conical and angular points. Trudy Moskov. Mat. Obšč.16, 209–292 (1967)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod 1969
Raviart, P-A: On the finite element approximation of nonlinear, problems. Computational methods in nonlinear mechanics (J.T. Oden ed.), pp. 413–425. Amsterdam: North-Holland 1980
Author information
Authors and Affiliations
Additional information
supported by the Fonds National Suisse de la Recherche Scientifique
Rights and permissions
About this article
Cite this article
Brezzi, F., Rappaz, J. & Raviart, P.A. Finite dimensional approximation of nonlinear problems. Numer. Math. 36, 1–25 (1980). https://doi.org/10.1007/BF01395985
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01395985