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Single step Galerkin approximations for parabolic problems. (English) Zbl 0378.65061


MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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References:

[1] J. BLAIR, Approximate Solution of Elliptic and Parabolic Boundary Value Problems, Thesis, University of California, Berkeley, 1970.
[2] J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), no. 137, 94 – 111. · Zbl 0353.65064
[3] J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), no. 2, 218 – 241. · Zbl 0364.65084 · doi:10.1137/0714015
[4] James H. Bramble and Vidar Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. (4) 101 (1974), 115 – 152. · Zbl 0306.65073 · doi:10.1007/BF02417101
[5] M. CROUZEIX, Sur l’Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta, Thesis, University of Paris VI, 1975.
[6] Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575 – 626. · Zbl 0224.35048 · doi:10.1137/0707048
[7] Todd Dupont, Some \?² error estimates for parabolic Galerkin methods, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 491 – 504.
[8] Hans-Peter Helfrich, Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen, Manuscripta Math. 13 (1974), 219 – 235 (German, with English summary). · Zbl 0323.65037 · doi:10.1007/BF01168227
[9] G. J. Makinson, Stable high order implicit methods for the numerical solution of systems of differential equations, Comput. J. 11 (1968/1969), 305 – 310. · Zbl 0167.15704 · doi:10.1093/comjnl/11.3.305
[10] Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937 – 958. · Zbl 0298.65071
[11] Syvert P. Nørsett, One-step methods of Hermite type for numerical integration of stiff systems, Nordisk Tidskr. Informationsbehandling 14 (1974), 63 – 77. · Zbl 0278.65078
[12] Vidar Thomée, Some convergence results for Galerkin methods for parabolic boundary value problems, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 55 – 88. Publication No. 33.
[13] Vidar Thomée, High order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), no. 139, 652 – 660. · Zbl 0367.65055
[14] 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
[15] Miloš Zlámal, Finite element methods for parabolic equations, Math. Comp. 28 (1974), 393 – 404.
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