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Sufficient conditions for the convergence, uniformly in ε, of a three point difference scheme for a singular perturbation problem

Part of the Lecture Notes in Mathematics book series (LNM,volume 679)

Abstract

We consider a general three point difference scheme (Ph) for a two point boundary value singular perturbation problem (P) with a parameter ε, which may be small, and without turning points. We give sufficient conditions for the convergence of the solution of (Ph) to that of (P) as h → 0 with order h, uniformly in ε. We state each step required in the derivation of this result, but we omit the detailed proof of each such step. We remark that, in particular, a well known scheme of Il’in fulfills these conditions and also that several widely used schemes are not convergent uniformly in ε.

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References

  1. A.M. Il’in, "Differencing scheme for a differential equation with a small parameter affecting the highest derivative", Math. Notes Acad.Sci. USSR 6 (1969), 596–602.

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  2. J.J.H. Miller, "Some finite difference schemes for a singular perturbation problem" in Constructive Function Theory. Proc.Int.Conf. Constructive Function Theory, Blagoevgrad 30 May–4 June 1977, Sofia (in print).

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© 1978 Springer-Verlag

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Miller, J.J.H. (1978). Sufficient conditions for the convergence, uniformly in ε, of a three point difference scheme for a singular perturbation problem. In: Ansorge, R., Törnig, W. (eds) Numerical Treatment of Differential Equations in Applications. Lecture Notes in Mathematics, vol 679. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067869

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  • DOI: https://doi.org/10.1007/BFb0067869

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08940-7

  • Online ISBN: 978-3-540-35715-5

  • eBook Packages: Springer Book Archive