Monoton einschliessende Iterationsverfahren for invers-isotone Diskretisierung nicht-linearer Zwei-Punkt-Randwertaufgaben zweiter Ordnung

Abstract

For discretization of equations −y″(s)+g(s, y(s), y′(s))=0, 0<s<1, with mixed boundary conditions and additional assumptions implying inverse isotonicity we analyze some iterative methods which do not require solving nonlinear equations at each step and which avoid using derivatives of g. Using Rheinboldt's (1970) theory of M-functions we prove existence and unicity of the solution of the discrete problem and then show that iterative methods of Jacobi and of Gauß-Seidel type are convergent and monotonic (i. e. have the monotonic inclusion property), the latter converging faster. A particular result of H. B. Keller (1966, 1968) is thus reobtained but without unnecessary restrictions on g, and in addition we have monotonicity. Convergence alone (without monotonicity) has also been shown by K. H. Müller (1971) who used the weak row-sum criterion for systems of nonlinear equations.

Wir stellen hier Ergebnisse aus der an der RWTH Aachen angefertigten Diplomarbeit des zweiten Verfassers dar.