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On the non-existence of $\varepsilon$-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems

Authors: Paul A. Farrell, John J. H. Miller, Eugene O’Riordan and Grigorii I. Shishkin
Journal: Math. Comp. 67 (1998), 603-617
MSC (1991): Primary 34B15, 65L12; Secondary 34L30, 65L10
MathSciNet review: 1451321
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Abstract: In this paper fitted finite difference methods on a uniform mesh with internodal spacing $h$, are considered for a singularly perturbed semilinear two-point boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge $\varepsilon$-uniformly in the maximum norm to the solution of the differential equation as the mesh spacing $h$ goes to zero. Numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors.

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  • 1. A. Brandt, I. Yavneh, Inadequacy of First-order Upwind Difference Schemes for some Recirculating Flows, J. Comput. Phys., 93, (1991) pp. 128-143. MR 91m:76075
  • 2. E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, Ireland, 1980. MR 82h:65053
  • 3. T.M. El-Mistikawy, M.J. Werle, Numerical Method for Boundary Layers with Blowing - the Exponential Box Scheme, AIAA J., 16 (1978), pp. 749-751.
  • 4. P.A. Farrell, Sufficient conditions for uniform convergence of a class of difference schemes for a singularly perturbed problem, IMA J. Numer. Anal., 7(4), (1987), pp 459-472. MR 90h:65130
  • 5. P.A. Farrell, E.C. Gartland, Jr., On the Scharfetter-Gummel Discretization for Drift-Diffusion Continuity Equations, in ``Computational Methods for Boundary and Interior Layers in Several Dimensions'', J.J.H. Miller, ed., pp. 51-79, Boole Press, Dublin, Ireland, (1991). MR 92k:65153
  • 6. P.A. Farrell, A. Hegarty, On the determination of the order of uniform convergence, in Proc. of $13^{\rm th}$ IMACS World Congress, Dublin, Ireland, 1991, pp. 501-502.
  • 7. P.A. Farrell, J.J.H. Miller, E. O'Riordan, G.I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation, SIAM J. Numer. Anal., 33, no. 3, (1996), pp. 1135-1149. MR 97b:65086
  • 8. A.M. Il'in, Difference scheme for a differential equation with a small parameter affecting the highest derivative, Mat. Zametki, 6 (1969), pp. 237-248. MR 41:4823
  • 9. V.D. Liseikin, On the numerical solution of second order equations with a small parameter affecting the highest derivatives, Chisl. Metody Mechaniki Splosh. Sredy, Novosibirsk, 14, n. 3 (1983), pp. 98-108. MR 86a:65075
  • 10. P.A. Markowich, C.A. Ringhofer, S. Selberherr, M. Lentini, A singular perturbation approach for the analysis of the fundamental semiconductor equations, IEEE Trans. Electron Devices, 30, n. 9 (1983), pp. 1165-1180.
  • 11. J.J.H. Miller, On the convergence, uniformly in $\varepsilon $, of difference schemes for a two-point boundary singular perturbation problem, in Numerical analysis of singular perturbation problems, P.W. Hemker, J.J.H. Miller, eds., Academic Press (1979), pp. 467-474. MR 81f:65061
  • 12. J.J.H. Miller, E.O'Riordan, G.I.Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996. CMP 97:10
  • 13. J.J.H. Miller, W. Song, A Tetrahedral Mixed Finite Element Method for the Stationary Semiconductor Continuity Equations SIAM J. Numer. Anal. , 31 n. 1 (1994), pp. 196-216.
  • 14. K.W.Morton, Numerical Solution of Convection Diffusion Problems, Chapman and Hall, London, 1996.
  • 15. J.J.H. Miller ed., Applications of Advanced Computational Methods for Boundary and Interior Layers, Boole Press, Dublin, Ireland, (1993) MR 94e:65006
  • 16. J.D. Murray, Lectures on Nonlinear Differential Equation Models in Biology, Clarendon Press, Oxford, 1977.
  • 17. Koichi Niijima, An error analysis for a difference scheme of exponential type applied to a nonlinear singular perturbation problem without turning points, J. Comput. Appl. Math., 15, no. 1, (1986) pp. 93-101. MR 87h:65129
  • 18. H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. - Convection-Diffusion and Flow Problems, Springer-Verlag, New-York, 1996.
  • 19. W.V. van Roosbroeck, Theory of flows of electrons and holes in germanium and other semiconductors, Bell Syst. Tech. J., 29 (1950), pp. 560-607.
  • 20. D.L. Scharfetter, H.K. Gummel, Large-signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices, 16, n. 1 (1969), pp. 64-77.
  • 21. G.I. Shishkin, Approximation of solutions of singularly perturbed boundary-value problems with a corner boundary layer, Comput. Maths. Math. Phys., 27, n. 5 (1987), pp. 54-63. (J. Vychisl. Mat. i Mat. Fiz., 27, n. 9 (1987), pp. 1360-1372.) MR 89a:65149
  • 22. G.I. Shishkin, Grid approximation of boundary value problems with regular boundary layer, Part 1, Part 2, Preprint INCA, 1990.
  • 23. G.I. Shishkin, Difference scheme for solving elliptic equations with small parameters affecting the derivatives, Banach Centre Publ., Warsaw, 3 (1978), pp. 89-92. MR 80d:65111
  • 24. G.I. Shishkin, Grid approximation of singularly perturbed boundary value problems with a regular boundary layer, Sov. J. Numer. Anal. Math. Modelling, 4, n. 5 (1989), pp. 397-417. MR 91b:65138
  • 25. Szollosi-Nagy, The Discretization of the Continuous Linear Cascade by Means of State Space Analysis J. Hydrol., 58, (1982) pp. 223-236.
  • 26. R. Vulanovi\'{c}, P.A. Farrell, P.Lin, Numerical Solution of Non-linear Singular Perturbation Problems Modeling Chemical Reactions, in ``Applications of Advanced Computational Methods for Boundary and Interior Layers'', J.J.H. Miller, ed., Boole Press, Dublin, Ireland, pp. 192-213 (1993) MR 95a:65119
  • 27. V. W. Weekman, Jr., R. L. Gorring, Influence of volume change on gas-phase reactions in porous catalysts, J. Catalysis 4 (1965), 260-270.

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Additional Information

Paul A. Farrell
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

John J. H. Miller
Affiliation: Department of Mathematics, Trinity College, Dublin 2, Ireland

Eugene O’Riordan
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

Grigorii I. Shishkin
Affiliation: Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

Keywords: Semilinear boundary value problem, singular perturbation, finite difference scheme, $\varepsilon$-uniform convergence, uniform mesh, frozen fitting factor
Received by editor(s): July 3, 1995
Received by editor(s) in revised form: February 9, 1996
Additional Notes: Supported in part under NSF grant DMS-9627244.
The first author was supported in part by the Research Council of Kent State University.
The fourth author was supported in part by the Russian Foundation for Basic Research under Grant N 95-01-00039.
Article copyright: © Copyright 1998 American Mathematical Society

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