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The gradient-finite element method for elliptic problems. (English) Zbl 0987.65121
Author’s summary: The coupling of the Sobolev space gradient method and the finite element method is developed. The Sobolev space gradient method reduces the solution of a quasilinear elliptic problem to a sequence of linear Poisson equations. These equations can be solved numerically by an appropriate finite element method. This coupling of the two methods will be called the gradient-finite element method (GFEM). Linear convergence of the GFEM is proved via suitable error control in the steps of the iteration. The GFEM defines an already preconditioned iteration in the sense that the theoretical ratio of convergence of the Sobolev space gradient method is preserved. Finally, a numerical example illustrates the method.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
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[1] Ciarlet, Ph., The finite element method for elliptic problems, (1978), North Holland Amsterdam
[2] Molchanov, I.N.; Nikolenko, L.D., Introduction to the finite element method, (1989), Naukova Dumka Kiev, (in Russian) · Zbl 0717.65078
[3] Axelsson, O., Iterative solution methods, (1994), Cambridge Univ. Press · Zbl 0795.65014
[4] Kantorovich, L.V.; Akilov, G.P., Functional analysis, (1982), Pergamon Press · Zbl 0484.46003
[5] Necas, J., Introduction to the theory of elliptic equations, (1986), J. Wiley and Sons · Zbl 0657.76058
[6] Gajewski, H.; Gröger, K.; Zacharias, K., Nichtlineare operatorgleichungen und operatordifferentialgleichungen, (1974), Akademie-Verlag Berlin · Zbl 0289.47029
[7] Karátson, J., The gradient method for a class of nonlinear operators in Hilbert space and applications to quasilinear differential equations, Pure math. appl., 6, 2, 191-201, (1995) · Zbl 0852.47035
[8] Simon, L.; Baderko, E., Linear partial differential equations of second order, (1983), Tankönyvkiadó Budapest, (in Hungarian)
[9] Hackbusch, W., Theorie und numerik elliptischer differentialgleichungen, (1986), Teubner Stuttgart · Zbl 0609.65065
[10] Kovács, I.; Lendvai, J.; Vörös, G., Effect of precipitation structure on the work hardening process, Materials sci. forum, 217-222, 1275-1280, (1996)
[11] Mikhlin, S.G., The numerical performance of variational methods, (1971), Walters-Noordhoff · Zbl 0209.18301
[12] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[13] Bornemann, F.A.; Erdmann, B.; Kornhuber, R., A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. numer. anal., 33, 3, 1188-1204, (1996) · Zbl 0863.65069
[14] Ihlenburg, F.; Babuška, I., Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Internat. J. numer. methods engrg., 38, 22, 3745-3774, (1995) · Zbl 0851.73062
[15] Ženíšek, A., Nonlinear elliptic and evolution problems and their finite element approximations, Computational mathematics and applications, (1990), Academic Press London · Zbl 0731.65090
[16] Szabó, B.; Babuška, I., Finite element analysis, (1991), J. Wiley and Sons
[17] Misovskych, I.P., Formulas of interpolation quadraturas, (1981), Nauka Moscow, (in Russian)
[18] Ciarlet, Ph., Basic error estimates for elliptic problems, () · Zbl 0875.65086
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