An analysis of a superconvergence result for a singularly perturbed boundary value problem
Authors:
Eugene O'Riordan and Martin Stynes
Journal:
Math. Comp. 46 (1986), 8192
MSC:
Primary 65L10
MathSciNet review:
815833
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Abstract: We give a new proof that the ElMistikawy and Werle finitedifference scheme is uniformly secondorder accurate for a nonselfadjoint singularly perturbed boundary value problem. To do this, we use exponential finite elements and a discretized Green's function. The proof is direct, gives the nodal errors explicitly in integral form, and involves much less computation than in previous proofs of the result.
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 A. E. Berger, J. M. Solomon & M. Ciment, "An analysis of a uniformly accurate difference method for a singular perturbation problem," Math. Comp., v. 37, 1981, pp. 7994. MR 616361 (83f:65121)
 [2]
 T. M. ELMistikawy & M. J. Werle, "Numerical method for boundary layers with blowingthe exponential box scheme," AIAA J., v. 16, 1978, pp. 749751.
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 P. P. N. De Groen & P. W. Hemker, "Error bounds for exponentially fitted Galerkin methods applied to stiff twopoint boundary value problems," in Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, New York, 1979, pp. 217249. MR 556520 (81a:65076)
 [4]
 A. F. Hegarty, J. J. H. Miller & E. O'Riordan, "Uniform second order difference schemes for singular perturbation problems," in Boundary and Interior LayersComputational and Asymptotic Methods (J. J. H. Miller, ed.), Boole Press, Dublin, 1980, pp. 301305. MR 589380 (83h:65095)
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 P. W. Hemker, A Numerical Study of Stiff Twopoint Boundary Value Problems, Mathematical Centre, Amsterdam, 1977.
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 A. M. Iĺin, "Differencing scheme for a differential equation with a small parameter affecting the highest derivative," Mat. Zametki, v. 6, 1969, pp. 237248; English transl, in Math. Notes, v. 6, 1969, pp. 596602. MR 0260195 (41:4823)
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 R. B. Kellogg & A. Tsan, "Analysis of some difference approximations for a singular perturbation problem without turning points," Math. Comp., v. 32, 1978, pp. 10251039. MR 0483484 (58:3485)
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 E. O'Riordan, Finite Element Methods for Singularly Perturbed Problems, Ph. D. thesis, School of Mathematics, Trinity College, Dublin, 1982.
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 E. O'Riordan, "Singularly perturbed finite element methods," Numer. Math., v. 44, 1984, pp. 425434. MR 757497 (85m:65080)
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 D. R. Smith, "The multivariable method in singular perturbation analysis," SIAM Rev., v. 17, 1975, pp. 221273. MR 0361331 (50:13776)
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 M. Stynes & E. O'Riordan, "A superconvergence result for a singularly perturbed boundary value problem," in BAIL III, Proc. Third International Conference on Boundary and Interior Layers (J. J. H. Miller, ed.), Boole Press, Dublin, 1984, pp. 309313. MR 774624 (86b:65084)
 [12]
 M. Stynes & E. O'Riordan, "A uniformly accurate finite element method for a singular perturbation problem in conservative form," SIAM J. Numer. Anal. (To appear.) MR 831623 (88a:65092)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158338
PII:
S 00255718(1986)08158338
Article copyright:
© Copyright 1986
American Mathematical Society
