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


Authors: Adam Andersson and Stig Larsson
Journal: Math. Comp. 85 (2016), 1335-1358
MSC (2010): Primary 65M60, 60H15, 60H35, 65C30
Published electronically: August 20, 2015
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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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Additional Information

Adam Andersson
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: adam.andersson@chalmers.se

Stig Larsson
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: stig@chalmers.se

DOI: https://doi.org/10.1090/mcom/3016
Keywords: Nonlinear stochastic heat equation, SPDE, finite element, error estimate, weak convergence, multiplicative noise, Malliavin calculus
Received by editor(s): July 26, 2013
Received by editor(s) in revised form: September 18, 2013, and November 18, 2014
Published electronically: August 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society