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

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Authors Adam Andersson
Stig Larsson
Publication year 2012
Published at Department of Mathematical Sciences
Department of Mathematical Sciences, Mathematics
Language en
Links arxiv.org/abs/1212.5564
https://gup.ub.gu.se/file/89222
Keywords non-linear 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 approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method. 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 only has a solution in one space dimension. In order to get results for higher dimensions, colored noise is considered here, besides the white noise case where considerably weaker assumptions on the noise term is needed. 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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