Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
Preprint, 2012

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.

error estimate

Malliavin calculus

non-linear heat equation

SPDE

multiplicative noise

weak convergence

finite element

Författare

Adam Andersson

Chalmers, Matematiska vetenskaper

Göteborgs universitet

Stig Larsson

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Fundament

Grundläggande vetenskaper