On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems

Abstract

A general theory is developed for calculating equidistributing meshes $\{ t_i \} $ for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0. \] It is shown that the original problem and the equidistribution constraints on the mesh $\{ t_i \} $ can be replaced by a transformed boundary-value problem on a uniform mesh. Existence, uniqueness, and convergence of Newton’s method for the discrete solution and the equidistributing mesh are proved. Equidistribution of arc length is given for boundary-layer problems. Five sample problems are solved with different methods of choosing the mesh $\{ t_i \} $.