An analysis of a uniformly convergent finite difference/finite element scheme for a model singularperturbation problem
Author:
Eugene C. Gartland
Journal:
Math. Comp. 51 (1988), 93106
MSC:
Primary 65L10; Secondary 65L60
MathSciNet review:
942145
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Abstract: Uniform convergence is proved for the ElMistikawyWerle discretization of the problem on (0,1), , , subject only to the conditions and . The principal tools used are a certain representation result for the solutions of such problems that is due to the author [Math. Comp., v. 48, 1987, pp. 551564] and the general stability results of Niederdrenk and Yserentant [Numer. Math., v. 41, 1983, pp. 223253]. Global uniform convergence is proved under slightly weaker assumptions for an equivalent PetrovGalerkin formulation.
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 [1]
 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]
 J. Douglas, Jr. & T. Dupont, "Some superconvergence results for Galerkin methods for the approximate solution of twopoint boundary value problems," in Topics in Numerical Analysis, Proc. Royal Irish Academy Conf., 1972 (J. J. H. Miller, ed.), pp. 8992. MR 0366044 (51:2295)
 [3]
 T. M. ElMistikawy & M. J. Werle, "Numerical method for boundary layers with blowingthe exponential box scheme," AIAA J., v. 16, 1978, pp. 749751.
 [4]
 E. C. Gartland, Jr., Strong Stability and a Representation Result for a Singular Perturbation Problem, Technical Report AMS 871, Dept. of Mathematics, Southern Methodist University, January, 1987.
 [5]
 E. C. Gartland, Jr., "Uniform highorder difference schemes for a singularly perturbed twopoint boundary value problem," Math. Comp., v. 48, 1987, pp. 551564. MR 878690 (89a:65116)
 [6]
 E. C. Gartland, Jr., An Analysis of the AllenSouthwell FiniteDifference Scheme for a Model Singular Perturbation Problem, Technical Report AMS 872, Dept. of Mathematics, Southern Methodist University, April, 1987.
 [7]
 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)
 [8]
 T. Kato, Perturbation Theory for Linear Operators, 2nd ed., SpringerVerlag, Berlin, 1980.
 [9]
 S. H. Leventhal, "An operator compact implicit method of exponential type," J. Comput. Phys., v. 46, 1982, pp. 138165. MR 665807 (84b:76007)
 [10]
 J. Lorenz, Stability and Consistency Analysis of Difference Methods for Singular Perturbation Problems, Proc. Conf. on Analytical and Numerical Approaches to Asymptotic Problems in Analysis, June 913, 1980, Univ. of Nijmegen, The Netherlands (O. Axelsson, L. Frank, and A. Van der Sluis, eds.), NorthHolland, Amsterdam, 1981. MR 605505 (83b:65077)
 [11]
 K. Niederdrenk & H. Yserentant, "Die gleichmässige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme," Numer. Math., v. 41, 1983, pp. 223253. MR 703123 (84j:65049)
 [12]
 E. O'Riordan & M. Stynes, "An analysis of a superconvergence result for a singularly perturbed boundary value problem," Math. Comp., v. 46, 1986, pp. 8192. MR 815833 (87b:65107)
 [13]
 M. H. Protter & H. P. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1967. MR 0219861 (36:2935)
 [14]
 D. R. Smith, A Green Function for a Singularly Perturbed Dirichlet Problem, Technical Report, Dept. of Mathematics, University of California, San Diego, March, 1984.
 [15]
 D. R. Smith, Singular Perturbation Theory, Cambridge Univ. Press, Cambridge, 1985. MR 812466 (87d:34001)
 [16]
 M. Stynes & E. O'Riordan, "A finite element method for a singularly perturbed boundary value problem," Numer. Math., v. 50, 1986, pp. 115. MR 864301 (88e:65101)
 [17]
 W. G. Szymczak & I. Babuška, "Adaptivity and error estimation for the finite element method applied to convection diffusion problems," SIAM J. Numer. Anal., v. 21, 1984, pp. 910954. MR 760625 (86c:65143)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809421456
PII:
S 00255718(1988)09421456
Article copyright:
© Copyright 1988
American Mathematical Society
