Advances in Computational Mathematics 6 (t996) 65-76 65 A collocation method for singularly perturbed two-point boundary value problems with splines in tension* Miljenko Marugid and Mladen Rogina Depalcment of Mathematics, University of Zagreb, Bijeni?ka 30, 10000 Zagreb, Croatia E-mail: Miljenko.Marusic @math.hr Received 15 December 1994; revised June 1996 Communicated by L.L. Schumaker An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problem ay" + by' + cy = f(.,y), y(0) = yo, y(1) = yj. Sharp error bounds tbr the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (lal = ~ << 1), the represen- tation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory. Keywords: Spline in tension, ordinary differential equation. AMS subject classification: 65D07, 65L10, 65L60, 76D30. 1. Introduction The aim is to prove the convergence of the collocation by spline in tension for the nonlinear boundary value problem ay"+by'+cy=y(.,y), y(O)=y0, y(1)=yt, (1) and give a choice of tension parameters for a singularly perturbed problem (1) (lal = c << 1), which makes the convergence at least linear, independently of the knot distribution. * Supported by grant 1-01-254 of the Ministry of Science and Technology, Croatia. © J.C. Baltzer AG, Science Publishers