| Title:
|
Strong convergence estimates for pseudospectral methods (English) |
| Author:
|
Heinrichs, Wilhelm |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
37 |
| Issue:
|
6 |
| Year:
|
1992 |
| Pages:
|
401-417 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Strong convergence estimates for pseudospectral methods applied to ordinary boundary value problems are derived. The results are also used for a convergence analysis of the Schwarz algorithm (a special domain decomposition technique). Different types of nodes (Chebyshev, Legendre nodes) are examined and compared. (English) |
| Keyword:
|
pseudospectral |
| Keyword:
|
collocation |
| Keyword:
|
Schwarz algorithm |
| Keyword:
|
strong convergence estimates |
| Keyword:
|
domain decomposition |
| Keyword:
|
Legendre nodes |
| Keyword:
|
Chebyshev nodes |
| MSC:
|
34B05 |
| MSC:
|
35J25 |
| MSC:
|
65L10 |
| MSC:
|
65L60 |
| MSC:
|
65N30 |
| MSC:
|
65N35 |
| idZBL:
|
Zbl 0767.65064 |
| idMR:
|
MR1185797 |
| DOI:
|
10.21136/AM.1992.104520 |
| . |
| Date available:
|
2008-05-20T18:44:20Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104520 |
| . |
| Reference:
|
[1] L. Brutman: On the Lebesgue function for polynomial interpolation.Siam J. Numer. Anal. 15 (1978), 694-704. Zbl 0391.41002, MR 0510554, 10.1137/0715046 |
| Reference:
|
[2] C. Canuto A. Quarteroni: Approximation result for orthogonal polynomials in Sobolev spaces.Math. Comput. 38 (1982), 67-86. MR 0637287, 10.1090/S0025-5718-1982-0637287-3 |
| Reference:
|
[3] C. Canute: Boundary conditions in Chebyshev and Legendre methods.Siam J. Numer. Anal. 23 (1986), 815-831. MR 0849284, 10.1137/0723052 |
| Reference:
|
[4] C. Canuto A. Quarteroni: Variational methods in the theoretical analysis of spectral approximations.in Spectral Methods for Partial Differential Equations , Society for Industrial and Applied Mathematics, Philadelphia, PA (1984), 55-78 (R. G. Voigt, D. Gottlieb and M. Y. Hussaini, eds.). MR 0758262 |
| Reference:
|
[5] C. Canuto A. Quarteroni: Spectral and pseudospectral methods for parabolic problems with nonperiodic boundary conditions.Calcolo 18 (1981), 197-218. MR 0647825, 10.1007/BF02576357 |
| Reference:
|
[6] C. Canuto D. Funaro: The Schwarz algorithm for spectral methods.Siam J. Numer. Anal. 25 (1988), 24-40. MR 0923923, 10.1137/0725003 |
| Reference:
|
[7] L. Collatz: Differentialgleichungen.Teubner Studienbucher, Stuttgart, 1973. Zbl 0267.65001, MR 0352575 |
| Reference:
|
[8] J. W. Cooley A. W. Lewis P. D. Walch: The Fast Transform Algorithm: Programming considerations in the calculation of sine, cosine and Laplace transform.J. Sound vib. 12 (1970), 105-112. |
| Reference:
|
[9] R. De Vore: On Jackson's theorem.J. Approx. Theory 1 (1968), 314-318. 10.1016/0021-9045(68)90008-7 |
| Reference:
|
[10] H. Ehlich K. Zeller: Auswertung der Normen von Interpolations-operatoren.Math. Analen 164 (1986), 105-112. MR 0194799 |
| Reference:
|
[11] L. W. Kantorowitsch G. P. Akilow: Funktionalanalysis in normierten Räumen.Akademie-Verlag, Berlin, 1964. MR 0177273 |
| Reference:
|
[12] I. P. Natanson: Constructive function theory. III. Interpolation and approximation quadratures.Frederick Ungar Publishing CO., New York, 1965. |
| Reference:
|
[13] M. J. Pоwel: On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria.Com. J. 9 (1967), 404-407. MR 0208807, 10.1093/comjnl/9.4.404 |
| Reference:
|
[14] T. J. Rivlin: The Lebesgue constants for polynomial interpolation.in Functional analysis and its application (H. G. Garnir et al., Springer-Verlag, ed.), Berlin-Heidelberg-New York, 1974, pp. 422-437. Zbl 0299.41005, MR 0399706 |
| Reference:
|
[15] G. Rodrigue P. Saylor: Inner/outer iterative methods and numerical Schwarz algorithm II.-Proceedings of the IBM Conference on Vector and Parallel Processors for Scientific Computations, Rome, 1985. MR 0825967 |
| Reference:
|
[16] G. Rodrigue J. Simon: A generalization of the numerical Schwarz algorithm., Computing Methods in Applied Sciences and Engineering VI (R. Glowinski and J. L. Lions, eds.), North Holland, 1984. MR 0806784 |
| Reference:
|
[17] H. A. Schwarz: Gesammelte Mathematische Abhandlungen, Vol. 2.Springer-Verlag, Berlin. |
| Reference:
|
[18] G. Szegö: Orthogonal polynomials.Am. Math. Soc., New York, 1939. |
| Reference:
|
[19] C. Temperton: On the FACR(1) algorithm for the discrete Poisson equation.J. Соmр. Phys. 34 (1980), 314-329. MR 0562366 |
| Reference:
|
[20] G. M. Vainikko: Differential Equations 1.(1965), 186-194. |
| Reference:
|
[21] G. M. Vainikko: The convergence of the collocation method for nonlinear differential equations.USSR Соmр. Math. and Math. Phys. 6 (1966), 47-58. MR 0196945, 10.1016/0041-5553(66)90031-0 |
| Reference:
|
[22] H. Werner R. Schaback: Praktische Mathematik II.Springer-Verlag, Berlin-Heidelberg- New York, 1972. MR 0520918 |
| Reference:
|
[23] K. Witsch: Konvergenzaussagen für Projektionsverfahren bei linearen Operatoren, insbesondere Randwertaufgaben.Doctoral Thesis, Köln, 1974. |
| Reference:
|
[24] K. Witsch: Konvergenzaussagen für Projektionsverfahren bei linearen Operatoren.Numer. Math. 27 (1977), 339-354. Zbl 0336.65031, MR 0443361, 10.1007/BF01396182 |
| Reference:
|
[25] T. A. Zang Y. S. Wong M. Y. Hussaini: Spectral multigrid methods for elliptic equations I.J. Соmр. Phys. 48 (1992), 485-501. MR 0755459 |
| Reference:
|
[26] T. A. Zang Y. S. Wong M. Y. Hussaini: Spectral multigrid methods for elliptic equations II.J. Соmр. Phys. 54 (1984), 489-507. MR 0755456 |
| . |
