On best possible order of convergence estimates in the collocation method and Galerkin's method for singularly perturbed boundary value problems for systems of first-order ordinary differential equations

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the L∞-norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal. Introduction It is well known that the spline collocation method for a nonstiff boundary value problem leads to a priori high-order accuracy estimates in the uniform norm [1]–[3]. For the Galerkin method in nonstiff problems the corresponding estimates are quasioptimal [4]–[6]. For the investigation of stiff systems it is appropriate to use strongly nonuniform meshes [12]–[15]. This circumstance significantly complicates the problem. Moreover, for stiff problems, it is difficult to select the principal part of a differential operator. To overcome these difficulties, the authors of [7]–[11] proposed Petrov-Galerkin type methods involving special bases in the test spaces; by means of them it may be possible to approximate solutions very well, not only in the center of an interval but also in boundary layers. In the present article we use these ideas. For numerical analysis we use C quadratic splines on meshes proposed by N. S. Bakhvalov. These meshes have a little number of knots, but they are denser and closer in the boundary layers. This allows us to obtain high-order accuracy with small additional computational work. The estimates obtained in this article have the same accuracy as analogous estimates for nonstiff boundary value problems. It is shown that these estimates are unimprovable, and, for the Galerkin method, they are quasioptimal. Note that for collocation methods similar ideas are used in the papers by Asher and Weiss [12]–[13] and by Ringhofer [14], but they use other meshes and splines of high defects. Received by the editor May 28, 1994 and, in revised form, February 11, 1995 and May 26, 1996. 1991 Mathematics Subject Classification. Primary 65-02, 65L99; Secondary 65G99, 45A10.