Abstract
The paper serves as a review on the basic results showing how functional analytic tools have been applied in numerical analysis. It deals with abstract Cauchy problems and present how their solutions are approximated by using space and time discretisations. To this end we introduce and apply the basic notions of operator semigroup theory. The convergence is analysed through the famous theorems of Trotter and Kato, Lax, and Chernoff. We also list some of their most important applications.
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References
Bátkai, A., Csomós, P., Nickel, G.: Operator splittings and spatial approximations for evolution equations. J. Evol. Equ. 9, 613–636 (2009)
Bátkai, A., Csomós, P., Farkas, B.: Operator splitting with spatial-temporal discretization. In: Arendt, W., Ball, J.A., Behrndt, J., Förster, K.-H., Mehrmann, V., Trunk, C. (eds.) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, 161–172. Springer, Basel (2012)
Brenner, P., Thomée, V.: On rational approximations of semigroups. SIAM J. Numer. Anal. 16, 683–694 (1979)
Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238–242 (1968)
Engel, K.-J.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics. Springer-Verlag, New York (2000)
Faragó, I., Havasi, Á.: On the convergence and local splitting error of different splitting schemes. Prog. Comput. Fluid Dyn. 5, 495–504 (2005)
Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)
Ito, K., Kappel, F.: Evolution Equations and Approximations. World Scientific, Singapore (2002)
Kato, T.: On the trotter-lie product formula. Proc. Japan Acad. 50, 694–698 (1974)
Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1976)
Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)
Palencia, C.: Stability of rational multistep approximations of holomorphic semigroups. Math. Comput. 64, 591–599 (1995)
Palencia, C., Sanz-Serna, J.M.: An extension of the lax-richtmyer theory. Numer. Math. 44, 279–283 (1984)
Pazy, A.: Semigroups of Linear Operator and Applications to Partial Differential Equations. Springer, New York (1983)
Trotter, H.F.: Approximation of semi-groups of operators. Pac. J. Math. 8, 887–919 (1958)
Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)
Acknowledgments
P. Csomós and I. Faragó kindly acknowledge the support of the bilateral Hungarian-Austrian Science and Technology program TET_10-1-2011-0728. I. Fekete was supported by the European Union and the State of Hungary, co-financed by the European Social Fund witihin the framework of TÁMOP-4.2.4.A/2-11/1-2012-0001 ‘National Program of Excellence’–convergence program.
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Csomós, P., Faragó, I., Fekete, I. (2015). Operator Semigroups for Convergence Analysis. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science(), vol 9045. Springer, Cham. https://doi.org/10.1007/978-3-319-20239-6_4
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DOI: https://doi.org/10.1007/978-3-319-20239-6_4
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