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Error estimates for a Galerkin approximation of a parabolic control problem. (English) Zbl 0434.65092


MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
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References:

[1] Babuška, I.; Aziz, A. K.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 3-359 (1972), New York: Academic Press, New York · Zbl 0268.65052
[2] Baker, G. A.; Bramble, J. H.; Thomée, V., Single step Galerkin approximations for parabolic problems, Math. Comp., 31, 140, 818-847 (1977) · Zbl 0378.65061 · doi:10.2307/2006116
[3] Ju. M. Berezanskii,Expansion in eigenfunctions of self adjoint operators, Naukova Duma, Kiev, 1965; English transl., Trans. Math. Monographs, vol.17, Amer. Math. Soc., Providence, R.I., 1968. · Zbl 0142.37203
[4] Bramble, J. H.; Schatz, A. H.; Thomée, V.; Wahlbin, L. B., Some convergence estimates for Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal., 14, 2, 218-241 (1977) · Zbl 0364.65084 · doi:10.1137/0714015
[5] Bramble, J. H.; Thomée, V., Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl., 101, 115-152 (1974) · Zbl 0306.65073 · doi:10.1007/BF02417101
[6] Bramble, J. H.; Thomée, V., Semidiscrete-least squares methods for a parabolic boundary value problem, Math. Comp., 26, 119, 633-647 (1972) · Zbl 0268.65060 · doi:10.2307/2005092
[7] Lions, J. L.; Magenes, E., Non homogeneous boundary value problems and applications, vol. I-II (1972), New York: Springer-Verlag, New York · Zbl 0223.35039
[8] Lions, J. L., Optimal control of systems governed by partial differential equations (1971), New York: Springer-Verlag, New York · Zbl 0203.09001
[9] J. C. Nedelec,Schémas d’approximations pour des équations intégre différentielles de Riccati, Thesis, University of Paris, 1970.
[10] Schechter, M., On L^v estimates and regularity II, Math. Scand., 13, 47-69 (1963) · Zbl 0131.09505
[11] Thomée, V.; De Boor, C., Some convergence results for Galerkin methods for parabolic boundary value problems, Mathematical Aspects of Finite Elements in Partial Differential Equations, 55-88 (1974), New York: Academic Press, New York · Zbl 0343.65046
[12] R. Winther,A numerical Galerkin method for a parabolic control problem, Ph. D. Thesis, Cornell University, 1977.
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