Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

作者：Brehier Charles Edouard^*; Kopec Marie

来源：IMA Journal of Numerical Analysis, 2017, 37(3): 1375-1410.

DOI：10.1093/imanum/drw030

• stochastic partial differential equations
• invariant measures and ergodicity
• weak approximation
• Euler scheme
• finite element method
• Poisson equation

We study the long-time behavior of fully discretized semilinear stochastic partial different equations (SPDEs) with additive space-time white noise, which admits a unique invariant probability measure mu. We show that the average of (regular) test functions with respect to the (possibly nonunique) invariant laws of the approximations are close to the corresponding average with respect to mu.
More precisely, we analyse the rate of convergence with respect to time and space discretization parameters. Here we focus on the discretization in time thanks to a scheme of Euler type, and on a finite element discretization in space. The main new contribution here is the treatment of the spatial error.
The technique of the proof is original in the SPDE context: we generalize the approach of Mattingly et al. (2010, Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal., 48, 552-577), which relies on the use of a Poisson equation, to an infinite-dimensional setting. We show that the rates of convergence for the invariant laws are given by the corresponding weak orders of the discretization on finite time intervals: order 1/2 with respect to the time step and order 1 with respect to the mesh size.

• 出版日期2017-7