Previous |  Up |  Next


pseudospectral; collocation; Schwarz algorithm; strong convergence estimates; domain decomposition; Legendre nodes; Chebyshev nodes
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.
[1] L. Brutman: On the Lebesgue function for polynomial interpolation. Siam J. Numer. Anal. 15 (1978), 694-704. DOI 10.1137/0715046 | MR 0510554 | Zbl 0391.41002
[2] C. Canuto A. Quarteroni: Approximation result for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982), 67-86. DOI 10.1090/S0025-5718-1982-0637287-3 | MR 0637287
[3] C. Canute: Boundary conditions in Chebyshev and Legendre methods. Siam J. Numer. Anal. 23 (1986), 815-831. DOI 10.1137/0723052 | MR 0849284
[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
[5] C. Canuto A. Quarteroni: Spectral and pseudospectral methods for parabolic problems with nonperiodic boundary conditions. Calcolo 18 (1981), 197-218. DOI 10.1007/BF02576357 | MR 0647825
[6] C. Canuto D. Funaro: The Schwarz algorithm for spectral methods. Siam J. Numer. Anal. 25 (1988), 24-40. DOI 10.1137/0725003 | MR 0923923
[7] L. Collatz: Differentialgleichungen. Teubner Studienbucher, Stuttgart, 1973. MR 0352575 | Zbl 0267.65001
[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.
[9] R. De Vore: On Jackson's theorem. J. Approx. Theory 1 (1968), 314-318. DOI 10.1016/0021-9045(68)90008-7
[10] H. Ehlich K. Zeller: Auswertung der Normen von Interpolations-operatoren. Math. Analen 164 (1986), 105-112. MR 0194799
[11] L. W. Kantorowitsch G. P. Akilow: Funktionalanalysis in normierten Räumen. Akademie-Verlag, Berlin, 1964. MR 0177273
[12] I. P. Natanson: Constructive function theory. III. Interpolation and approximation quadratures. Frederick Ungar Publishing CO., New York, 1965.
[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. DOI 10.1093/comjnl/9.4.404 | MR 0208807
[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. MR 0399706 | Zbl 0299.41005
[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
[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
[17] H. A. Schwarz: Gesammelte Mathematische Abhandlungen, Vol. 2. Springer-Verlag, Berlin.
[18] G. Szegö: Orthogonal polynomials. Am. Math. Soc., New York, 1939.
[19] C. Temperton: On the FACR(1) algorithm for the discrete Poisson equation. J. Соmр. Phys. 34 (1980), 314-329. MR 0562366
[20] G. M. Vainikko: Differential Equations 1. (1965), 186-194.
[21] G. M. Vainikko: The convergence of the collocation method for nonlinear differential equations. USSR Соmр. Math. and Math. Phys. 6 (1966), 47-58. DOI 10.1016/0041-5553(66)90031-0 | MR 0196945
[22] H. Werner R. Schaback: Praktische Mathematik II. Springer-Verlag, Berlin-Heidelberg- New York, 1972. MR 0520918
[23] K. Witsch: Konvergenzaussagen für Projektionsverfahren bei linearen Operatoren, insbesondere Randwertaufgaben. Doctoral Thesis, Köln, 1974.
[24] K. Witsch: Konvergenzaussagen für Projektionsverfahren bei linearen Operatoren. Numer. Math. 27 (1977), 339-354. DOI 10.1007/BF01396182 | MR 0443361 | Zbl 0336.65031
[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
[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
Partner of
EuDML logo