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Time-discretised Galerkin approximations of parabolic stochastic PDE's

Published online by Cambridge University Press:  17 April 2009

W. Grecksch  and
P.E. Kloeden
W. Grecksch
Affiliation:
Institut für Optimierung und Stochastik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Hale (Saale), Germany.
P.E. Kloeden
Affiliation:
School of Computing and Mathematics, Deakin University, Geelong Campus, Geelong Vic. 3217, Australia.
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Abstract

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The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above by

where [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 1 , August 1996 , pp. 79 - 85
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Prato, G. Da and Zabczyk, G., Stochastic equations in infinite dimensions (Cambridge University Press, Cambridge, 1992).CrossRefGoogle Scholar
[2]Grecksch, W. and Tudor, C., Stochastic evolution equations (Akademie—Verlag, Berlin, 1995).Google Scholar
[3]Kloeden, P.E. and Platen, E., Numerical solution of stochastic differential equations (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar