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The Trotter-Kato theorem and approximation of PDEs. (English) Zbl 0893.47025

Summary: We present formulations of the Trotter-Kato theorem for approximation of linear C\({}_0\)-semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes’ equation as illustrative examples.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47H05 Monotone operators and generalizations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35L99 Hyperbolic equations and hyperbolic systems
65J10 Numerical solutions to equations with linear operators
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[1] H. T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control Theory Adv. Tech. 4 (1988), no. 1, 73 – 90.
[2] J. H. Bramble, A. H. Schatz, V. ThomĂ©e, 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
[3] R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21 (1990), no. 2, 374 – 393. · Zbl 0688.65080
[4] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077
[5] K. Ito and F. Kappel, A uniformly differentiable approximation scheme for delay systems using splines, Appl. Math. Optim. 23 (1991), no. 3, 217 – 262. · Zbl 0738.65055
[6] Kazufumi Ito, Franz Kappel, and Dietmar Salamon, A variational approach to approximation of delay systems, Differential Integral Equations 4 (1991), no. 1, 51 – 72. · Zbl 0754.34082
[7] Kazufumi Ito and Janos Turi, Numerical methods for a class of singular integro-differential equations based on semigroup approximation, SIAM J. Numer. Anal. 28 (1991), no. 6, 1698 – 1722. · Zbl 0744.65103
[8] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009
[9] I. Lasiecka and A. Manitius, Differentiability and convergence rates of approximating semigroups for retarded functional-differential equations, SIAM J. Numer. Anal. 25 (1988), no. 4, 883 – 907. · Zbl 0654.65054
[10] Seymour V. Parter, On the roles of ”stability” and ”convergence” in semidiscrete projection methods for initial-value problems, Math. Comp. 34 (1980), no. 149, 127 – 154. · Zbl 0424.65044
[11] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[12] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. · Zbl 0155.47502
[13] Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. · Zbl 0417.35003
[14] Roger Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Studies in Mathematics and its Applications, Vol. 2. · Zbl 0383.35057
[15] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887 – 919. · Zbl 0099.10302
[16] Wolf von Wahl, The equations of Navier-Stokes and abstract parabolic equations, Aspects of Mathematics, E8, Friedr. Vieweg & Sohn, Braunschweig, 1985. · Zbl 0575.35074
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