Gradient-finite element method for nonlinear Neumann problems

• Autorzy: Faragó I. , Karátson J.
• Języki publikacji: EN

Abstrakty

EN

We consider the numerical solution of quasilinear elliptic Neumann problems. The basic difficulty is the non-injectivity of the operator, which can be overcome by suitable factorization. We extend the gradient-finite element method (GFEM), introduced earlier by the authors for Dirichlet problems, to the Neumann problem. The algorithm is constructed and its convergence is proved.

Słowa kluczowe

EN

Neumann boundary value problems; gradient-finite element method; non-injective nonlinear operator; factorization

PL

problemy granicznych wartości Neumann`a; gradient metoda elementów skończonych; nieliniowy, nie-iniektywny operator; rozkład na czynniki

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2001
• Tom: Vol. 7, nr 2
• Strony: 257--269
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Faragó I.

• Eötvös Lorând University, Department of Applied Analysis, H-1518 Budapest, pf. 120, Hungary

autor

Karátson J.

• Eötvös Lorând University, Department of Applied Analysis, H-1518 Budapest, pf. 120, Hungary

Bibliografia

• [1] Axelsson, O., Karat.son J., Double Sobolev gradient preconditioning for elliptic problems, Report 0016, Dept. Math., Univ. Nijmegen, April 2000 (submitted to Numer. Methods Partial Differential Equations).
• [2] 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) (1996), 1188-1204.
• [3] Egorov, Yu.V., Shubin, M.A., Encyclopedia of Mathematical Sciences, Partial Differential Equations /, Springer, Berlin, 1992.
• [4] Faierman, M., Regularity of solutions of an elliptic BVP in a rectangle, Comm. Partial Differential Equations 12 (1987), 285-305.
• [5] Faragó, I., Karatson, J., The gradient-finite element method for elliptic problems, Comput. Math. Appl. 42 (2001), 1043-1053.
• [6] Gajewski, H., Gröger, K., Zacharias. K.. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
• [7] Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
• [8] Hackbusch, W., Theorie und Numerik elliptischer Differentialgleichungen, Teubner, Stuttgart, 1986.
• [9] Gill, D., Tadmor. E., An 0(N2) method for computing the eigensystem of N x N symmetric tridiagonal matrices by the divide and conquer approach, SIAM J. Sei. Comput. 11(1) (1990), 161-173.
• [10] Hsiao, G. C., A modified Galerkin scheme for elliptic equations with natural boundary conditions, in: “Numerical Mathematics and Applications”, IMACS Trans. Sei. Comput. 85, I, North-Holland, Amsterdam-New York, 1986, 193-197.
• [11] Kantorovich, L.V., Akilov, G.P., Functional Analysis, Pergamon Press, Oxford- Elmstad, 1982.
• [12] Karátson, J., The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations, J. Appl. Anal. 3(2) (1997), 225-237.
• [13] Karátson, J., Gradient method for non-injective operators in Hilbert space with application to Neumann problems, Appl. Math. 26(3) (1999), 333-346.
• [14] Kovács, I., Lendvai, J., Vörös, G., Effect of precipitation structure on the work hardening process, Materials Sei. Forum 217-222 (1996), 1275-1280.
• [15] Křížek, M., Neittaanmäki, P., Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1996.
• [16] Lieberman., G.M., The conormal derivative problem for equations of variational type in nonsmooth domains, TVans. Amer. Math. Soc., 330 (1992), 41-67.
• [17] Neuberger, J. W., Sobolev Gradients and Differential Equations, Lecture Notes in Math. 1670. Springer-Verlag, Berlin, 1997.
• [18] Strang, G., Fix, G.J., An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, 1973.
• [19] Szabó, B., Babuśka, I., Finite Element Analysis, J.Wiley and Sons, New York, 1991.
• [20] Vainberg, M., Variational Method and the Method of Monotone Operators in the Theory of Nonlinear Equations, J. Wiley and Sons, New York-Toronto, 1973.
• [21] Ženíšek, A., Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations, Comput. Math. Appl., Academic Press, Inc., London, 1990.