×

zbMATH — the first resource for mathematics

Multistep-Galerkin methods for hyperbolic equations. (English) Zbl 0417.65057

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35G10 Initial value problems for linear higher-order PDEs
35G15 Boundary value problems for linear higher-order PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Garth A. Baker, A finite element method for first order hyperbolic equations, Math. Comput. 29 (1975), no. 132, 995 – 1006. · Zbl 0323.65035
[2] G. A. BAKER & J. H. BRAMBLE, Semidiscrete and Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations, Rep. No. 22, Centre de Mathématiques Appliqueés de l’Ecole Polytechnique, Paris, 1977.
[3] M. CROUZEIX, Sur l’Approximation des Équations Différentielles Opérationelles Linéaires par des Méthodes de Runge-Kutta, Thèse de Doctorat d’Etat, Université Paris VI, 1975.
[4] J. E. Dendy, Two methods of Galerkin type achieving optimum \?² rates of convergence for first order hyperbolics, SIAM J. Numer. Anal. 11 (1974), 637 – 653. · Zbl 0293.65077 · doi:10.1137/0711052 · doi.org
[5] V. A. DOUGALIS, High Order Fully Discrete Galerkin Approximations to Hyperbolic Equations, Ph. D. Thesis, Harvard University, May 1976.
[6] Todd Dupont, \?²-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880 – 889. · Zbl 0239.65087 · doi:10.1137/0710073 · doi.org
[7] Todd Dupont, Galerkin methods for first order hyperbolics: an example, SIAM J. Numer. Anal. 10 (1973), 890 – 899. · Zbl 0237.65070 · doi:10.1137/0710074 · doi.org
[8] G. Fix and N. Nassif, On finite element approximations to time-dependent problems, Numer. Math. 19 (1972), 127 – 135. · Zbl 0244.65063 · doi:10.1007/BF01402523 · doi.org
[9] E. Gekeler, Linear multistep methods and Galerkin procedures for initial boundary value problems, SIAM J. Numer. Anal. 13 (1976), no. 4, 536 – 548. · Zbl 0335.65042 · doi:10.1137/0713046 · doi.org
[10] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. · Zbl 0112.34901
[11] J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. · Zbl 0258.65069
[12] P. Lesaint, Finite element methods for symmetric hyperbolic equations, Numer. Math. 21 (1973/74), 244 – 255. · Zbl 0283.65061 · doi:10.1007/BF01436628 · doi.org
[13] J. L. LIONS & E. MAGENES, Problèmes aux Limites Non Homogènes et Applications, Vols. I & II, Dunod, Paris, 1968. · Zbl 0165.10801
[14] M. S. Mock, Projection methods with different trial and test spaces, Math. Comp. 30 (1976), no. 135, 400 – 416. · Zbl 0337.65057
[15] M. S. Mock, Explicit finite element schemes for first order symmetric hyperbolic systems, Numer. Math. 26 (1976), no. 4, 367 – 378. · Zbl 0345.65045 · doi:10.1007/BF01409959 · doi.org
[16] I. J. SCHOENBERG, ”Contributions to the problem of approximation of equidistant data by analytic functions,” Parts A and B, Quart. Appl. Math., v. 4, 1946, pp. 45-99, 112-141.
[17] Gilbert Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547 – 583.
[18] Blair Swartz and Burton Wendroff, The relative efficiency of finite difference and finite element methods. I. Hyperbolic problems and splines, SIAM J. Numer. Anal. 11 (1974), 979 – 993. · Zbl 0294.65055 · doi:10.1137/0711076 · doi.org
[19] Vidar Thomée, Convergence estimates for semi-discrete Galerkin methods for initial-value problems, Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1972) Springer, Berlin, 1973, pp. 243 – 262. Lecture Notes in Math., Vol. 333.
[20] Vidar Thomée and Burton Wendroff, Convergence estimates for Galerkin methods for variable coefficient initial value problems, SIAM J. Numer. Anal. 11 (1974), 1059 – 1068. · Zbl 0292.65052 · doi:10.1137/0711081 · doi.org
[21] Lars Wahlbin, A modified Galerkin procedure with Hermite cubics for hyperbolic problems, Math. Comput. 29 (1975), no. 132, 978 – 984. · Zbl 0323.65034
[22] B. WENDROFF, On Finite Elements for Equations of Evolution, Rep. LA-DC-72-1220, Los Alamos, N. M., 1972.
[23] Mary Fanett Wheeler, A priori \?\(_{2}\) error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723 – 759. · Zbl 0232.35060 · doi:10.1137/0710062 · doi.org
[24] Miloš Zlámal, Finite element multistep discretizations of parabolic boundary value problems, Math. Comp. 29 (1975), 350 – 359.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.