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Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

Journal article
Authors Adam Andersson
Stig Larsson
Published in Mathematics of Computation
Volume 85
Pages 1335-1358
ISSN 0025-5718
Publication year 2016
Published at Department of Mathematical Sciences, Mathematics
Pages 1335-1358
Language en
Links dx.doi.org/10.1090/mcom/3016
Keywords Nonlinear stochastic heat equation, SPDE, finite element, error estimate, weak convergence, multiplicative noise, Malliavin calculus
Subject categories Numerical analysis, Mathematical statistics

Abstract

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.

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