Galerkin approximations for the stochastic Burgers equation

  • Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated. These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including the stochastic heat equation, stochastic reaction diffusion equations and the stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations. The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the infinite dimensional SPDE uniformly in space, instead of the usual Hilbert-space estimates, that were shown before.

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Author:Dirk BlömkerORCiDGND, Arnulf Jentzen
Frontdoor URL
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2009-22)
Year of first Publication:2009
Publishing Institution:Universität Augsburg
Contributing Corporation:Johann Wolfgang Goethe-University, Institute of Mathematics, Frankfurt am Main
Release Date:2009/09/22
GND-Keyword:Nichtlineare partielle Differentialgleichung; Stochastische partielle Differentialgleichung; Burgers-Gleichung; Galerkin-Methode
Erschienen in SIAM Journal on Numerical Analysis, 51, 1, S. 694-715,
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand