| Title:
|
$N$-widths for singularly perturbed problems (English) |
| Author:
|
Stynes, Martin |
| Author:
|
Kellogg, R. Bruce |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
343-352 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Kolmogorov $N$-widths are an approximation theory concept that, for a given problem, yields information about the optimal rate of convergence attainable by any numerical method applied to that problem. We survey sharp bounds recently obtained for the $N$-widths of certain singularly perturbed convection-diffusion and reaction-diffusion boundary value problems. (English) |
| Keyword:
|
$N$-width |
| Keyword:
|
singularly perturbed |
| Keyword:
|
differential equation |
| Keyword:
|
boundary value problem |
| Keyword:
|
convection-diffusion |
| Keyword:
|
reaction-diffusion |
| MSC:
|
34E15 |
| MSC:
|
35B25 |
| MSC:
|
35K57 |
| MSC:
|
41A46 |
| MSC:
|
65L10 |
| MSC:
|
65L20 |
| MSC:
|
65N15 |
| idZBL:
|
Zbl 1005.41009 |
| idMR:
|
MR1981538 |
| DOI:
|
10.21136/MB.2002.134155 |
| . |
| Date available:
|
2012-10-05T13:07:41Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134155 |
| . |
| Reference:
|
[1] R. A. Adams: Sobolev Spaces.Academic Press, New York, 1975. Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] J.-P. Aubin: Approximation of Elliptic Boundary-Value Problems.Wiley Interscience, New York, 1972. Zbl 0248.65063, MR 0478662 |
| Reference:
|
[3] J. Bergh, J. Löfström: Interpolation Spaces.Springer, Berlin, 1976. MR 0482275 |
| Reference:
|
[4] P. Grisvard: Elliptic problems in nonsmooth domains.Pitman, Boston, 1985. Zbl 0695.35060, MR 0775683 |
| Reference:
|
[5] R. B. Kellogg, M. Stynes: Optimal approximability of solutions of singularly perturbed two-point boundary value problems.SIAM J. Numer. Anal. 34 (1997), 1808–1816. MR 1472198, 10.1137/S0036142995290269 |
| Reference:
|
[6] R. B. Kellogg, M. Stynes: $N$-widths and singularly perturbed boundary value problems.SIAM J. Numer. Anal. 36 (1999), 1604–1620. MR 1706743, 10.1137/S0036142997327257 |
| Reference:
|
[7] R. B. Kellogg, M. Stynes: $N$-widths and singularly perturbed boundary value problems II.SIAM J. Numer. Anal. 39 (2001), 690–707. MR 1860257, 10.1137/S0036142900371489 |
| Reference:
|
[8] G. G. Lorentz: Approximation of Functions.2nd edition, Chelsea Publishing Company, New York, 1986. Zbl 0643.41001, MR 0917270 |
| Reference:
|
[9] J. M. Melenk: On $N$-widths for elliptic problems.J. Math. Anal. Appl. 247 (2000), 272–289. Zbl 0963.35047, MR 1766938, 10.1006/jmaa.2000.6862 |
| Reference:
|
[10] J. T. Oden, J. N. Reddy: An Introduction to the Mathematical Theory of Finite Elements.Wiley-Interscience, New York, 1976. MR 0461950 |
| Reference:
|
[11] A. Pinkus: $N$-Widths in Approximation Theory.Springer, Berlin, 1985. Zbl 0551.41001, MR 0774404 |
| Reference:
|
[12] H.-G. Roos, M. Stynes, L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations.Springer, Berlin, 1996. MR 1477665 |
| Reference:
|
[13] G. Sun, M. Stynes: Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems I: reaction-diffusion-type problems.IMA J. Numer. Anal. 15 (1995), 117–139. MR 1311341, 10.1093/imanum/15.1.117 |
| . |
