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Optimal B-spline collocation method for self-adjoint singularly perturbed boundary value problems. (English) Zbl 1119.65064

The authors propose a numerical method based on a non-overlapping domain decomposition by using B-spline for singularly perturbed two-point boundary-value problems of the form

$-{\epsilon }^{2}{u}^{\text{'}\text{'}}\left(x\right)+b\left(x\right)u\left(x\right)=f\left(x\right),\phantom{\rule{1.em}{0ex}}\left(0,1\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)={\rho }_{1},\phantom{\rule{1.em}{0ex}}u\left(1\right)={\rho }_{2}·$

Recently, R. K. Bawa and S. Natesan in [Comput. Math. Appl. 50, No. 8–9, 1371–1382 (2005; Zbl 1084.65070)] devised a non-overlapping domain decomposition method for the singular perturbation problem stated above by using quintic spline.

In this article, the authors follow the same domain decomposition idea proposed by Bawa and Natesan [loc. cit.], and use B-splines instead of quintic splines. Error estimates are derived for the B-spline scheme following the same steps for the quintic spline scheme as given in Bawa and Natesan [loc. cit.]. Also, the numerical examples given here do not reflect any improvement of the proposed minor change suggested by the authors as the error estimates are of same order as obtained by Bawa and Natesan [loc. cit.].

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B05 Linear boundary value problems for ODE 34E15 Asymptotic singular perturbations, general theory (ODE) 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 65L70 Error bounds (numerical methods for ODE)