The exponential dichotomy on general approximation scheme

V. Pastor ; S. Piskarev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 103-127.

Résumé

This paper is devoted to the numerical analysis of abstract parabolic problems $u'(t)=Au(t)$, $u(0)=u^0$ with hyperbolic generator $A$. We develop a general approach to establish a discrete dichotomy in a very general setting in the case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to initial value problems with exponentially decaying solutions in opposite time directions. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for the finite element method as well as finite difference methods.

Bibliographie

[1] Akhmerov R. R., Kamenskiĭ M. I., Potapov A. S., Rodkina A. E., Sadovskiĭ B. N., Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, 55, Birkhäuser, Basel, 1992 | DOI | MR | Zbl

[2] Appell J., “Measures of noncompactness, condensing operators and fixed points: An application-oriented survey”, Fixed Point Theory, 6:2 (2005), 157–229 | MR | Zbl

[3] Ashyralyev A., Sobolevskii P. E., Well-Posedness of Parabolic Difference Equations, Operator Theory: Advances and Applications, 69, Birkhäuser, Basel, 1994 | MR

[4] Banas J., Goebel K., Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., 60, Marcel Dekker, New York, 1980 | MR | Zbl

[5] Baskakov A. G., “Linear differential operators with unbounded operator coefficients, and semigroups of difference operators”, Mat. Zametki, 59:6 (1996), 811–820 | DOI | MR | Zbl

[6] Baskakov A. G., “On differential and difference Fredholm operators”, Dokl. Math., 76:2 (2007), 669–672 | DOI | MR | Zbl

[7] Baskakov A. G., Pastukhov A. I., “Spectral analysis of a weighted shift operator with unbounded operator coefficients”, Sib. Mat. Zh., 42:6 (2001), 1231–1243 | MR | Zbl

[8] Baskakov A. G., Sintyaev Yu. N., “Finite-difference operators in the study of differential operators: Solution estimates”, J. Differ. Equ., 46:2 (2010), 214–223 | DOI | MR | Zbl

[9] Beyn W.-J., “Numerical methods for dynamical systems”, Advances in Numerical Analysis (Lancaster, 1990), v. I, Oxford Sci. Publ., Oxford Univ. Press, Oxford, 1991, 175–236 | MR

[10] Beyn W.-J., Piskarev S., “Shadowing for discrete approximations of abstract parabolic equations”, Discrete Contin. Dynam. Systems Ser. B, 10:1 (2008), 19–42 | DOI | MR | Zbl

[11] Carvalho A. N., Piskarev S., “A general approximation scheme for attractors of abstract parabolic problems”, Numer. Funct. Anal. Optim., 27:7–8 (2006), 785–829 | DOI | MR | Zbl

[12] Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts Math., 194, Springer, Berlin, 2000 | MR | Zbl

[13] Garay B. M., “On structural stability of ordinary differential equations with respect to discretization methods”, Numer. Math., 72:4 (1996), 449–479 | DOI | MR | Zbl

[14] Garay B. M., Lee K., “Attractors under discretization with variable stepsize”, Discrete Contin. Dynam. Systems, 13:3 (2005), 827–841 | DOI | MR | Zbl

[15] Gohberg I. C., Krein M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, 1969 | MR | Zbl

[16] Grigorieff R. D., “Diskrete Approximation von Eigenwertproblemen. II. Konvergenzordnung”, Numer. Math., 24:5 (1975), 415–433 | DOI | MR | Zbl

[17] Guidetti D., Karasozen B., Piskarev S., “Approximation of abstract differential equations”, J. Math. Sci., 122:2 (2004), 3013–3054 | DOI | MR | Zbl

[18] Hale J. K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs, 25, Amer. Math. Soc., Providence, 1989 | MR

[19] Henry D., Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981 | MR

[20] Kaashoek M. A., Verduyn Lunel S. M., “An integrability condition on the resolvent for hyperbolicity of the semigroup”, J. Differ. Equ., 112:2 (1994), 374–406 | DOI | MR | Zbl

[21] Kato T., Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995 | MR

[22] Krasnosel'skij M. A., Zabrejko P. P., Geometrical Methods of Nonlinear Analysis, Grundlehren Math. Wiss., 263, Springer, Berlin, 1984 | DOI | MR | Zbl

[23] Krein S. G., Linear Differential Equations in Banach Space, Transl. Math. Monogr., 29, Amer. Math. Soc., Providence, 1971 | MR

[24] Larsson S., “Numerical analysis of semilinear parabolic problems”, The Graduate Student's Guide to Numerical Analysis' 98, Lect. Notes from the 8th EPSRC Summer School in Numerical Analysis (Leicester, GB, July 5–17, 1998), Comput. Math., 26, ed. M. Ainsworth, Springer, Berlin, 1999, 83–117 | MR | Zbl

[25] Larsson S., Sanz-Serna J. M., “The behavior of finite element solutions of semilinear parabolic problems near stationary points”, SIAM J. Numer. Anal., 31:4 (1994), 1000–1018 | DOI | MR | Zbl

[26] Larsson S., Sanz-Serna J. M., “A shadowing result with applications to finite element approximation of reaction-diffusion equations”, Math. Comput., 68:225 (1999), 55–72 | DOI | MR | Zbl

[27] Latushkin Y., Pogan A., Schnaubelt R., “Dichotomy and Fredholm properties of evolution equations”, J. Operator Theory, 58:2 (2007), 387–414 | MR | Zbl

[28] Ostermann A., Palencia C., “Shadowing for nonautonomous parabolic problems with applications to long-time error bounds”, SIAM J. Numer. Anal., 37:5 (2000), 1399–1419 | DOI | MR | Zbl

[29] Palmer K. J., “A perturbation theorem for exponential dichotomies”, Proc. Roy. Soc. Edinburgh Sect. A, 106:1–2 (1987), 25–37 | DOI | MR | Zbl

[30] Palmer K. J., “Exponential dichotomies and Fredholm operators”, Proc. Am. Math. Soc., 104:1 (1988), 149–156 | DOI | MR | Zbl

[31] Pilyugin S. Yu., Shadowing in Dynamical Systems, Lect. Notes Math., 1706, Springer, Berlin, 1999 | MR | Zbl

[32] Piskarev S., “Error estimates for approximation of semigroups of operators by Pade's fractions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1979, no. 4(203), 33–38 | MR | Zbl

[33] Piskarev S., “On approximation of holomorphic semigroups”, Tartu Riikl. Ül. Toimetised, 492 (1979), 3–23 | MR

[34] Piskarev S., Differential Equations in Banach Space and Their Approximation, Izd. Mosk. Univ., Moscow, 2005 (in Russian)

[35] Robinson R. C., An Introduction to Dynamical Systems: Continuous and Discrete, Pearson Prentice Hall, Upper Saddle River, 2004 | MR | Zbl

[36] Rottmann J., Spectral properties of mixed hyperbolic-parabolic systems, Diplomarbeit, Univ. of Bielefeld, 2005

[37] Sobolevskii P. E., “The theory of semigroups and the stability of difference schemes”, Operator Theory in Function Spaces, Proc. School (Novosibirsk, 1975), Nauka, Novosibirsk, 1977, 304–337 | MR

[38] Sell G. R., You Y., Dynamics of Evolutionary Equations, Appl. Math. Sci., 143, Springer, New York, 2002 | DOI | MR | Zbl

[39] Stuart A., “Convergence and stability in the numerical approximation of dynamical systems”, The State of the Art in Numerical Analysis (York, 1996), Inst. Math. Appl. Conf. Ser. New Ser., 63, Oxford Univ. Press, Oxford, 1997, 145–169 | MR | Zbl

[40] Stummel F., “Diskrete Konvergenz linearer Operatoren. III”, Linear Operators and Approximation, Proc. Conf. (Oberwolfach, 1971), Internat. Ser. Numer. Math., 20, Birkhäuser, Basel, 1972, 196–216 | MR

[41] Trotter H., “Approximation of semigroups of operators”, Pacific J. Math., 8 (1958), 887–919 | DOI | MR | Zbl

[42] Vainikko G., Funktionalanalysis der Diskretisierungsmethoden, Teubner-Texte zur Mathematik, Teubner, Leipzig, 1976 | MR | Zbl

[43] Vainikko G., “Approximative methods for nonlinear equations (two approaches to the convergence problem)”, Nonlinear Anal., 2 (1978), 647–687 | DOI | MR | Zbl

[44] Vasil'ev V. V., Piskarev S. I., “Differential equations in Banach spaces. II. Theory of cosine operator functions”, J. Math. Sci., 122:2 (2004), 3055–3174 | DOI | MR | Zbl

[45] Vu Q. P., “On the exponential stability and dichotomy of $C_0$-semigroups”, Studia Math., 132:2 (1999), 141–149 | MR | Zbl

[46] Vu Q. P., “The spectral radius, hyperbolic operators and Lyapunov's theorem”, Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lect. Notes Pure Appl. Math., 215, Marcel Dekker, New York, 2001, 187–194 | MR | Zbl

[47] Vu Q. P., “A new proof and generalizations of Gearhart's theorem”, Proc. Am. Math. Soc., 135:7 (2007), 2065–2072 | DOI | MR | Zbl