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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 $$-\varepsilon^2 u''(x) + b(x)u(x) = f(x), \quad (0,1), \quad u(0) = \rho_1, \quad u(1) = \rho_2. $$ Recently, {\it R. K. Bawa} and {\it 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.].

65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
34E15Asymptotic singular perturbations, general theory (ODE)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L70Error bounds (numerical methods for ODE)
Full Text: DOI
[1] Shishkin, G. I.: A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous boundary condition. Zh. vychisl. Mat. i mat. Fiz. 28, No. 11, 1649-1662 (1988) · Zbl 0662.65086
[2] Miller, J. J. H.; O’riordan, E.; Shishkin, G. I.: Fitted numerical methods for singular perturbation problems. (1996)
[3] Boglaev, I. P.: A variational difference scheme for a boundary value problem with a small parameter multiplying the highest derivative. Zh. vychisl. Mat. i mat. Fiz. 21, No. 4, 887-896 (1981) · Zbl 0495.65037
[4] Schatz, A. H.; Wahlbin, L. B.: On the finite element method for singularly perturbed reaction -- diffusion problems in two and one dimensions. Math. comp. 40, No. 161, 47-89 (1983) · Zbl 0518.65080
[5] Miller, J. J. H.: On the convergence, uniformly in &z.epsiv;, of difference schemes for a two point boundary singular perturbation problem. Proc. conf., math. Inst., catholic univ., nijmegen, 1978, 467-474 (1979)
[6] O’riordan, E.: Singularly perturbed finite element methods. Numer. math. 44, 425-434 (1984) · Zbl 0569.65065
[7] O’riordan, E.; Stynes, M.: A uniformly accurate finite element method for a singularly perturbed one-dimensional reaction-diffusion problem. Math. comput. 47, 555-570 (1986) · Zbl 0625.65073
[8] O’riordan, E.; Stynes, M.: An analysis of a super convergence result for a singularly perturbed boundary value problem. Math. comput. 46, 81-92 (1986) · Zbl 0612.65043
[9] Stynes, M.; O’riordan, E.: Uniformly accurate finite element method for a singularly perturbed problem in conservative form. SIAM J. Num. anal. 23, 369-375 (1986) · Zbl 0595.65091
[10] Stojanović, M.: Spline collocation method for singular perturbation problem. Glas. mat. Ser. III 37, No. 572/2, 393-403 (2002) · Zbl 1030.65086
[11] K. Surla, V. Jerković, Some possibilities of applying spline collocations to singular perturbation problems, in: Numerical methods and approximation theory, II (Novi Sad, 1985), pp. 19 -- 25. Univ. Novi Sad, Novi Sad, 1985.
[12] Kadalbajoo, M. K.; Aggarwal, V. K.: Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems. Appl. math. Comput. 161, No. 3, 973-987 (2005) · Zbl 1073.65062
[13] Bawa, R. K.; Natesan, S.: A computational method for self-adjoint singular perturbation problems using quintic spline. Comput. math. Appl. 50, No. 8-9, 1371-1382 (2005) · Zbl 1084.65070
[14] Ahlberg, J. H.; Nilson, E. N.; Walsh, J. L.: The theory of splines and their applications. (1967) · Zbl 0158.15901
[15] Swartz, B. K.; Varga, R. S.: Error bounds for spline and l-spline interpolation. J. approximation theory 6, 1-49 (1972) · Zbl 0242.41008
[16] Lucas, T. R.: Error bounds for interpolating cubic splines under various end conditions. SIAM J. Numer. anal. 11, 569-584 (1974) · Zbl 0286.65004
[17] Prenter, P. M.: Splines and variational methods. (1989) · Zbl 0718.65053
[18] Doolan, E. P.; Miller, J. J. H.; Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers. (1980) · Zbl 0459.65058
[19] Varah, J. M.: A lower bound for the smallest singular value of a matrix. Linear algebra appl. 11, 3-5 (1975) · Zbl 0312.65028
[20] Farrell, P. A.; Hegarty, A. F.; Miller, J. J. H.; O’riordan, E.; Shishkin, G. I.: Robust computational techniques for boundary layers. Applied mathematics (Boca raton) 16 (2000) · Zbl 0964.65083