Abstract
We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.
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Mikhlin S. G., “A remark on the Ritz method,” Dokl. Akad. Nauk SSSR, 106, No.3, 391–394 (1956).
Vorovich I. I., “On some direct methods in the nonlinear theory of oscillations of shallow shells,” Izv. Akad. Nauk SSSR Ser. Mat., 21, No.6, 747–784 (1957).
Lions J.-L., Some Methods for Solving Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).
Lions J.-L. and Magenes E., Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).
Ladyzhenskaya O. A., Boundary Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).
Zhelezovskaya L. A., Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., Kirichenko V. F., and Krys’ko V. A., “Convergence rate of the Bubnov-Galerkin method for hyperbolic equations,” Differentsial’nye Uravneniya, 26, No.2, 323–333 (1990).
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., “The Bubnov-Galerkin method for an abstract quasilinear problem on stationary action,” Differentsial’nye Uravneniya, 31, No.7, 1222–1231 (1995).
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., “Rate of convergence of the Galerkin method for one class of quasilinear evolution problems,” submitted to VINITI on January 8, 1998; No. 24-B98.
Strang G. and Fix G. J., An Analysis of the Finite Element Method [Russian translation], Mir, Moscow (1977).
Marchuk G. I. and Agoshkov V. I., An Introduction to Projection-Grid Methods [in Russian], Nauka, Moscow (1981).
Dendy J. E., Jr., “An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems,” SIAM J. Numer. Anal., 12, No.4, 541–565 (1975).
Dendy J. E., Jr., “Galerkin’s method for some highly nonlinear problems,” SIAM J. Numer. Anal., 14, No.2, 327–347 (1977).
Geveci T., “On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms,” Math. Comput., 42, No.166, 393–415 (1984).
Kok B. and Geveci T., “The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation,” Math. Comput., 44, No.170, 379–390 (1985).
Zlotnik A. A. “Estimates for the rate of convergence of the projection-grid methods for second-order hyperbolic equations,” in: Computational Processes and Systems [in Russian], Nauka, Moscow, 1991, No. 8, pp. 116–167.
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., “Estimates for the rate of convergence of the Galerkin method for abstract hyperbolic equations,” Mat. Zametki, 69, No.2, 223–234 (2001).
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E. and Bukesova N. N., “The error estimates of the projection method for an abstract quasilinear hyperbolic equation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 94–96 (1999).
Lyashko A. D. and Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., “Well-posedness of an operator-differential scheme and justification of the Galerkin method for hyperbolic equations,” Sibirsk. Zh. Vychisl. Mat., 3, No.4, 357–368 (2000).
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E. and Lyashko A. D., “The error estimates of the Galerkin method for quasilinear hyperbolic equations,” Differentsial’nye Uravneniya, 37, No.7, 941–949 (2001).
Morozov N. F., “Study of oscillations of a prismatic rod under transverse load,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 121–125 (1965).
Zhelezovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E., “On the existence and uniqueness of a solution and on the convergence rate of the Bubnov-Galerkin method for a quasilinear evolution problem in Hilbert space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 37–45 (1998).
Kre \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) n S. G., Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).
Kolmogorov A. N. and Fomin S. V., Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1981).
Akhiezer N. I., Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).
Beckenbach E. F. and Bellman R., Inequalities [Russian translation], Mir, Moscow (1965).
Gajewski H., Gröger K., and Zacharias K., Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978).
Lyusternik L. A. and Sobolev V. I., Elements of Functional Analysis [in Russian], Nauka, Moscow (1965).
Ciarlet P. G., The Finite Element Method for Elliptic Problems [Russian translation], Mir, Moscow (1980).
Mikhlin S. G., Numerical Realization of Variational Methods [in Russian], Nauka, Moscow (1966).
Dzhiskariani A. V., “On rapidity of the convergence of the Bubnov-Galerkin method,” Zh. Vychisl. Mat. i Mat. Fiziki, 4, No.2, 343–348 (1964).
Va \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) nikko G. M., “Some error estimates of the Bubnov-Galerkin method. I: Asymptotic estimates,” Uchen. Zap. Tartu Univ., No. 150, 188–201 (1964).
Zarubin A. G., “Convergence rate of the Faedo-Galerkin method for quasilinear nonstationary operator equations,” Differentsial’nye Uravneniya, 26, No.12, 2051–2059 (1990).
Smagin V. V., “The error estimates of semidiscrete approximations by Galerkin for parabolic equations with boundary condition of Neumann type,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 50–57 (1996).
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Original Russian Text Copyright © 2005 Zhelezovski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) S. E.
Translated from Sibirski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Matematicheski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Zhurnal, Vol. 46, No. 2, pp. 374–389, March–April, 2005.
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Zhelezovskii, S.E. On error estimates in the Galerkin method for hyperbolic equations. Sib Math J 46, 293–304 (2005). https://doi.org/10.1007/s11202-005-0030-1
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DOI: https://doi.org/10.1007/s11202-005-0030-1