Summary
We continue here the study of a general method of approximation of nonlinear equations in a Banach space yet considered in [2]. In this paper, we give fairly general approximation results for the solutions in a neighborhood of a simple limit point. We the apply the previous analysis to the study of Galerkin approximations for a class of variationally posed nonlinear problems and to a mixed finite element method for the NavierStokes equations.
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References
Berger, M.S.: Nonlinearity and functional analysis. New York: Academic Press, 1977
Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math.36, 1–25 (1980)
Fujii, H., Yamaguti, M.: Structure of singularities and its numerical realization in nonlinear elasticity. J. Math. Kyoto (in press) (1981).
Girault, V., Raviart, P.-A.: An analysis of a mixed finite element method for the Navier-Stokes equations. Numer. Math.33, 235–271 (1979)
Grisvard, P.: Singularité des solutions du problème de Stokes dans un polygone. Publication de I'Université de Nice (1978)
Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of bifurcation theory. (P.H. Rabinowitz ed.), New York: Academic Press, pp. 359–384, 1977
Kikuchi, F.: Numerical analysis of the finite element method applied to bifurcation problems of turning point type. ISAS Report no 564, University of Tokyo, 217–246 (1978)
Kikuchi, F.: Finite element approximations to bifurcation problems of turning point type. Theoretical and Applied Mechanics27, 99–144 (1979)
Kikuchi, F.: Finite element approximation to bifurcation problems of turning point type. Troisième colloque international sur les méthodes de calcul scientifique et technique (Versailles 1977). Berlin Heidelberg New York: Springer 1981 (in press)
Kondrat'ev, V.A.: Boundary value problems for elliptic equations in domains with canical and angular points. Trudy. Moskov. Mat. Obšč. 209–292 (1967)
Paumier, J.C.: Une méthode numérique pour le calcul des points de retournement. Application au problème de Dirichlet −Δu=λe u (in press) (1981)
Scholz, R.: Finite element turning points of the Navier-Stokes equations. Conference on Progress in the Theory and Practice of the Finite Element Method. Chalmers University of Technology, Göteborg (1979)
Simpson, R.B.: Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems. Math. Comput.26, 359–375 (1972)
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This work has been completed during a visit at the Université Pierre et Marie Curic and at the Ecole Polytechnique
Supported by the Fonds National Suisse de la Recherche Scientifique
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Brezzi, F., Rappaz, J. & Raviart, P.A. Finite dimensional approximation of nonlinear problems. Numer. Math. 37, 1–28 (1981). https://doi.org/10.1007/BF01396184
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DOI: https://doi.org/10.1007/BF01396184