On contracting interval iteration for nonlinear problems in Rn: part II—applications

Abstract

governs classes of evolution-type transport processes in the theories of diffusion, conduction of heat, chemical kinetics, etc. Explicit solutions of (1.1) with boundary conditions are known only in a few exceptional cases (e.g., [4]). Therefore, finite-dimensional approximations are of considerable practical interest. According to the equivalence theorem (e.g., [7]), the convergence of a sequence of such approximations is ensured if the conditions of consistency and stability are satisfied. Gershgorin’s method is a general approach for establishing the stability property in the case of nonlinear BVP (boundary value problems). This method transfers the property of inverse monotonicity from the BVP to the finite-dimensional approximation, where it has to be shown that this property holds uniformly as a grid parameter, m E N, tends to infinity. If A is an inverse monotone operator, then the operator inequality Au d Aw (making use of a partial ordering of the space U* on which A operates) implies that u d w for every pair of elements U, w E U* such that Au < Aw. The uniformity of the property of inverse monotonicity can be shown by use of available sufficient conditions if the classes of admissible functions c, k, and C$ in (1.1) are rather severely restricted. In addition, for fixed m, the qualitative property of convergence does not yield a quantitative estimate of the relationship between the local and the global discretization error [13, p. 131, where local error refers to the order of consistency of the difference scheme and the global error, for any fured m, is the distance between the restriction to the grid of the solution of the BVP and the solution of the difference scheme. Rather than verifying the qualitative properties of stability and convergence, for fmed m, here a quantitative error estimate is derived which relates the local to the global discretization error; this estimate depends on both the grid parameter m and the number of the iteration cycle of the numerical method to be employed. For the deduction of this relationship, it is sufficient that the finite system is individually (i.e., not necessarily uniformly) inverse monotone for each value of the grid parameter m E N.