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.


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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.