A uniformly accurate spline collocation method for a normalized flux

Authors:
Katarina Surla;Zorica Uzelac
Affiliations:
Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovic´a 4, 21000 Novi Sad, Yugoslavia;Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovic´a 6, 21000 Novi Sad Yugoslavia
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
Year:
2004

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Abstract

We are concerned with a two-point boundary value problem for a semilinear singularly perturbed reaction-diffusion equation with a singular perturbation parameter ε. Our goal is to construct global ε-uniform approximations of the solution y(x) and the normalized flux P(x) = ε(d/dx)y(x), using the collocation with the classical quadratic splines u(x) ∈ C1(I) on a slightly modified piecewise uniform mesh of Shishkin type. The constructed approximate solution and normalized flux converge ε-uniformly with the rate O(n-2 ln2 n) and O(n-1 ln n), respectively, on the Shishkin-type mesh, and with O(n-1 ln-2 n) and O(ln-3 n) when the mesh has to be modified. We present numerical experiments in support of these results.