Previous |  Up |  Next


diffusion problems; iterative solution; Banach fixed-point theorem; nonlinear heat-conduction; generalized Sobolev spaces of vector valued function
The present paper deals with the numerical solution of the nonlinear heat equation. An iterative method is suggested in which the iterations are obtained by solving linear heat equation. The convergence of the method is proved under very natural conditions on given input data of the original problem. Further, questions of convergence of the Galerkin method applied to the original equation as well as to the linear equations in the above mentioned iterative method are studied.
[1] J. R. Cannon, A. Fasano: Boundary value multidimensional problems in fast chemical reactions. Arch. Rat. Mech. Anal. 53 (1973), 1 - 13. DOI 10.1007/BF00735697 | MR 0348269 | Zbl 0276.35061
[2] H. Gajewski, K. Gröger: Ein Iterationsverfahren für Gleichungen mit einem maximal monotonen und einem stark monotonen Lipschitzstegigen Operator. Math. Nachr. 69 (1975)307-317. MR 0500342
[3] H. Gajewski, K. Kröger: Zur Konvergenz eines Itrationsverfahrens für Gleichungen der Form $Au' + Lu = f$. Math. Nachr. 69 (1975) 319-331. DOI 10.1002/mana.19750690129 | MR 0513156
[4] H. Gajewski, K. Gröger: Ein Projektions-Iterationsverfahren für Gleichungen der Form $Au' + Lu = f$. Math. Nachr. 73 (1976) 249-267. DOI 10.1002/mana.19760730119 | MR 0438697 | Zbl 0352.65024
[5] H. Gajewski K. Grüger K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Berlin 1974.
[6] Y. Konishi: On the nonlinear semigroups associated with $u_t = \Delta \beta(u)$ and $\beta (u_t) = \Delta u$. J. Math. Soc. Japan, 25 (1973) 622-628. MR 0326517
[7] W. Strauss: Evolution equations non-linear in the time derivative. J. Math. Mech., 15 (1966) 49-82. MR 0190807 | Zbl 0138.40001
Partner of
EuDML logo