| Title:
|
Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers (English) |
| Author:
|
Roos, Hans-Görg |
| Author:
|
Stynes, Martin |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
41 |
| Issue:
|
4 |
| Year:
|
1996 |
| Pages:
|
269-280 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Singularly perturbed problems of convection-diffusion type cannot be solved numerically in a completely satisfactory manner by standard numerical methods. This indicates the need for robust or $\epsilon $-uniform methods. In this paper we derive new conditions for such schemes with special emphasize to parabolic layers. (English) |
| Keyword:
|
numerical analysis |
| Keyword:
|
convection-diffusion problems |
| Keyword:
|
boundary layers |
| Keyword:
|
uniform convergence |
| MSC:
|
65N06 |
| MSC:
|
65N12 |
| idZBL:
|
Zbl 0870.65091 |
| idMR:
|
MR1395686 |
| DOI:
|
10.21136/AM.1996.134326 |
| . |
| Date available:
|
2009-09-22T17:51:39Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134326 |
| . |
| Reference:
|
[AS64] Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions.National Bureau of Standards, 1964. |
| Reference:
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[Ec73] Eckhaus, W.: Matched asymptotic expansions and singular perturbations.North-Holland, Amsterdam, 1973. Zbl 0255.34002, MR 0670800 |
| Reference:
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[Em73] Emel’janov, K.V.: A difference scheme for a three-dimensional elliptic equation with a small parameter multiplying the highest derivative.Boundary value problems for equations of mathematical physics, USSR Academy of Sciences, Ural Scientific Centre, 1973, pp. 30–42. (Russian) |
| Reference:
|
[Gu93] Guo, W.: Uniformly convergent finite element methods for singularly perturbed parabolic problems.Ph.D. Dissertation, National University of Ireland, 1993. |
| Reference:
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[HK90] Han, H., Kellogg, R.B.: Differentiability properties of solutions of the equation $-\epsilon ^2\Delta u+ru=f(x,y)$ in a square.SIAM J. Math. Anal., 21 (1990), 394–408. MR 1038899, 10.1137/0521022 |
| Reference:
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| Reference:
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| Reference:
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[Ro85] Roos, H.-G.: Necessary convergence conditions for upwind schemes in the two-dimensional case.Int. J. Numer. Meth. Eng. 21 (1985), 1459–1469. Zbl 0578.65098, MR 0799066, 10.1002/nme.1620210808 |
| Reference:
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[SK87] Shih, S.D., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem.SIAM J. Math. Anal., 18 (1987), 1467–1511. MR 0902346, 10.1137/0518107 |
| Reference:
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[Sh89] Shishkin, G.I.: Approximation of the solutions of singularly perturbed boundary-value problems with a parabolic boundary layer.U.S.S.R. Comput. Maths. Math. Physics 29 (1989), 1–10. Zbl 0709.65073, MR 1011021, 10.1016/0041-5553(89)90109-2 |
| Reference:
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[Si90] Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems with convective terms.Sov. J. Numer. Anal. Math. Modelling 5 (1990), 173–187. Zbl 0816.65051, MR 1122367, 10.1515/rnam.1990.5.2.173 |
| Reference:
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[Si92] Shishkin, G.I.: Methods of constructing grid approximations for singularly perturbed boundary value problems.Sov. J. Numer. Anal. Math. Modelling 7 (1992), 537–562. Zbl 0816.65072, MR 1202653 |
| Reference:
|
[ST92] Stynes, M., Tobiska, L.: Necessary $L_2$-uniform conditions for difference schemes for two-dimensional convection-diffusion problems.Computers Math. Applic. 29 (1995), 45–53. MR 1321058, 10.1016/0898-1221(94)00237-F |
| Reference:
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[Ys83] Yserentant, H.: Die maximale Konsistenzordnung von Differenzapproximationen nichtnegativer Art.Numer. Math. 42 (1983), 119–123. MR 0716478, 10.1007/BF01400922 |
| . |
