On the nonexistence of uniform finite difference methods on uniform meshes for semilinear twopoint boundary value problems
Authors:
Paul A. Farrell, John J. H. Miller, Eugene O’Riordan and Grigorii I. Shishkin
Journal:
Math. Comp. 67 (1998), 603617
MSC (1991):
Primary 34B15, 65L12; Secondary 34L30, 65L10
MathSciNet review:
1451321
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper fitted finite difference methods on a uniform mesh with internodal spacing , are considered for a singularly perturbed semilinear twopoint boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge uniformly in the maximum norm to the solution of the differential equation as the mesh spacing goes to zero. Numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors.
 1.
A.
Brandt and I.
Yavneh, Inadequacy of firstorder upwind difference schemes for
some recirculating flows, J. Comput. Phys. 93 (1991),
no. 1, 128–143. MR 1097117
(91m:76075), http://dx.doi.org/10.1016/00219991(91)90076W
 2.
E.
P. Doolan, J.
J. H. Miller, and W.
H. A. Schilders, Uniform numerical methods for problems with
initial and boundary layers, Boole Press, Dún Laoghaire, 1980.
MR 610605
(82h:65053)
 3.
T.M. ElMistikawy, M.J. Werle, Numerical Method for Boundary Layers with Blowing  the Exponential Box Scheme, AIAA J., 16 (1978), pp. 749751.
 4.
Paul
A. Farrell, Sufficient conditions for uniform convergence of a
class of difference schemes for a singularly perturbed problem, IMA J.
Numer. Anal. 7 (1987), no. 4, 459–472. MR 968518
(90h:65130), http://dx.doi.org/10.1093/imanum/7.4.459
 5.
Paul
A. Farrell and Eugene
C. Gartland Jr., On the ScharfetterGummel discretization for
driftdiffusion continuity equations, Computational methods for
boundary and interior layers in several dimensions, Adv. Comput. Methods
Bound. Inter. Layers, vol. 1, Boole, Dublin, 1991,
pp. 51–79. MR 1151840
(92k:65153)
 6.
P.A. Farrell, A. Hegarty, On the determination of the order of uniform convergence, in Proc. of IMACS World Congress, Dublin, Ireland, 1991, pp. 501502.
 7.
Paul
A. Farrell, John
J. H. Miller, Eugene
O’Riordan, and Grigori
I. Shishkin, A uniformly convergent finite difference scheme for a
singularly perturbed semilinear equation, SIAM J. Numer. Anal.
33 (1996), no. 3, 1135–1149. MR 1393906
(97b:65086), http://dx.doi.org/10.1137/0733056
 8.
A.
M. Il′in, A difference scheme for a differential equation
with a small parameter multiplying the highest derivative, Mat.
Zametki 6 (1969), 237–248 (Russian). MR 0260195
(41 #4823)
 9.
V.
D. Liseĭkin, Numerical solution of secondorder equations
with a small parameter multiplying the highest derivatives, Chisl.
Metody Mekh. Sploshn. Sredy 14 (1983), no. 3,
98–108 (Russian). MR 756927
(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. 11651180.
 11.
John
J. H. Miller, On the convergence, uniformly in 𝜖, of
difference schemes for a two point boundary singular perturbation
problem, Numerical analysis of singular perturbation problems (Proc.
Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press,
LondonNew York, 1979, pp. 467–474. MR 556537
(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. 196216.
 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, Advanced Computational Methods for
Boundary and Interior Layers, 2, Boole Press, Dublin, 1993. MR 1245729
(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 (1986), no. 1,
93–101. MR
843714 (87h:65129), http://dx.doi.org/10.1016/03770427(86)902414
 18.
H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations.  ConvectionDiffusion and Flow Problems, SpringerVerlag, NewYork, 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. 560607.
 20.
D.L. Scharfetter, H.K. Gummel, Largesignal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices, 16, n. 1 (1969), pp. 6477.
 21.
G.
I. Shishkin, Approximation of solutions of singularly perturbed
boundary value problems with a corner boundary layer, Zh. Vychisl.
Mat. i Mat. Fiz. 27 (1987), no. 9, 1360–1374,
1438 (Russian). MR 918127
(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. Šiškin, A difference scheme for the solution of
elliptic equations with small parameters multiplying the derivatives,
Mathematical models and numerical methods (Papers, Fifth Semester, Stefan
Banach Internat. Math. Center, Warsaw, 1975) Banach Center Publ.,
vol. 3, PWN, Warsaw, 1978, pp. 89–92 (Russian). MR 514372
(80d:65111)
 24.
G.
I. Shishkin, Grid approximation of singularly perturbed boundary
value problems with a regular boundary layer, Soviet J. Numer. Anal.
Math. Modelling 4 (1989), no. 5, 397–417. MR 1026911
(91b:65138)
 25.
SzollosiNagy, The Discretization of the Continuous Linear Cascade by Means of State Space Analysis J. Hydrol., 58, (1982) pp. 223236.
 26.
R.
Vulanović, Paul
A. Farrell, and P.
Lin, Numerical solution of nonlinear singular perturbation problems
modelling chemical reactions, Applications of advanced computational
methods for boundary and interior layers, Adv. Comput. Methods Bound.
Inter. Layers, vol. 2, Boole, Dublin, 1993, pp. 192–213. MR 1245738
(95a:65119)
 27.
V. W. Weekman, Jr., R. L. Gorring, Influence of volume change on gasphase reactions in porous catalysts, J. Catalysis 4 (1965), 260270.
 1.
 A. Brandt, I. Yavneh, Inadequacy of Firstorder Upwind Difference Schemes for some Recirculating Flows, J. Comput. Phys., 93, (1991) pp. 128143. 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. ElMistikawy, M.J. Werle, Numerical Method for Boundary Layers with Blowing  the Exponential Box Scheme, AIAA J., 16 (1978), pp. 749751.
 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 459472. MR 90h:65130
 5.
 P.A. Farrell, E.C. Gartland, Jr., On the ScharfetterGummel Discretization for DriftDiffusion Continuity Equations, in ``Computational Methods for Boundary and Interior Layers in Several Dimensions'', J.J.H. Miller, ed., pp. 5179, 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 IMACS World Congress, Dublin, Ireland, 1991, pp. 501502.
 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. 11351149. 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. 237248. 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. 98108. 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. 11651180.
 11.
 J.J.H. Miller, On the convergence, uniformly in , of difference schemes for a twopoint boundary singular perturbation problem, in Numerical analysis of singular perturbation problems, P.W. Hemker, J.J.H. Miller, eds., Academic Press (1979), pp. 467474. 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. 196216.
 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. 93101. MR 87h:65129
 18.
 H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations.  ConvectionDiffusion and Flow Problems, SpringerVerlag, NewYork, 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. 560607.
 20.
 D.L. Scharfetter, H.K. Gummel, Largesignal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices, 16, n. 1 (1969), pp. 6477.
 21.
 G.I. Shishkin, Approximation of solutions of singularly perturbed boundaryvalue problems with a corner boundary layer, Comput. Maths. Math. Phys., 27, n. 5 (1987), pp. 5463. (J. Vychisl. Mat. i Mat. Fiz., 27, n. 9 (1987), pp. 13601372.) 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. 8992. 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. 397417. MR 91b:65138
 25.
 SzollosiNagy, The Discretization of the Continuous Linear Cascade by Means of State Space Analysis J. Hydrol., 58, (1982) pp. 223236.
 26.
 R. Vulanovi\'{c}, P.A. Farrell, P.Lin, Numerical Solution of Nonlinear 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. 192213 (1993) MR 95a:65119
 27.
 V. W. Weekman, Jr., R. L. Gorring, Influence of volume change on gasphase reactions in porous catalysts, J. Catalysis 4 (1965), 260270.
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
34B15,
65L12,
34L30,
65L10
Retrieve articles in all journals
with MSC (1991):
34B15,
65L12,
34L30,
65L10
Additional Information
Paul A. Farrell
Affiliation:
Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Email:
farrell@mcs.kent.edu
John J. H. Miller
Affiliation:
Department of Mathematics, Trinity College, Dublin 2, Ireland
Email:
jmiller@tcd.ie
Eugene O’Riordan
Affiliation:
School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email:
oriordae@ccmail.dcu.ie
Grigorii I. Shishkin
Affiliation:
Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia
Email:
grigorii@shishkin.ural.ru
DOI:
http://dx.doi.org/10.1090/S0025571898009223
PII:
S 00255718(98)009223
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 DMS9627244.
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 950100039.
Article copyright:
© Copyright 1998
American Mathematical Society
