Galerkin Approximations for the Stochastic Burgers Equation

Abstract

Abstract. Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated.These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation and a stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations.The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the solution of the infinite dimensional SPDE uniformly in space, i.e., in the L ∞ -topology, instead of the usual Hilbert space estimates in the L 2 -topology, that were shown before.Key words. Galerkin approximations, stochastic partial differential equation, stochastic heat equation, stochastic reaction diffusion equation, stochastic Burgers equation, strong error criteria. AMS subject classifications. 60H15, 35K901. Introduction. In this work we present a general abstract result for the spatial approximation of stochastic evolution equations with additive noise via Galerkin methods. This abstract result is applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation and a stochastic Burgers equation. In all examples we need to verify the following conditions. First, we need the rate of approximation of the linear equation obtained by omitting the nonlinear term in the stochastic evolution equation. Then one needs a quite weak Lipschitz condition for the nonlinearity and finally a uniform bound on the sequence of approximations. These results are the key for the main theorem (see Theorem 3.1). The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the solution of the infinite dimensional SPDE uniformly in space, i.e., in the L ∞ -topology, instead of the usual Hilbert space estimates shown before in the L 2 -topology. Although there are several different methods using finite dimensional approximations like, for instance, spectral Galerkin, finite elements, or wavelets, we focus here on the spectral Galerkin method. Thus the finite dimensional approximations are given by an expansion in terms of the eigenfunctions of a dominant linear operator. This spectral Galerkin method is one of the key tools in the analysis of stochastic or deterministic PDEs. For SPDEs see, for example, [16,9,17,2], where the Galerkin method was used to establish the existence of solutions. Moreover, spectral methods are an effective tool for numerical simulations, especially on domains, like the interval,

Full discretization of the stochastic Burgers equation with correlated noise

“…The main novelty in this article is to estimate the spatial and temporal discretization error in the L ∞ -topology in the case of colored noise. This is different from the usual space-time white noise, that was considered before in [2] for spatial discretization.…”

“…for every t ∈ [0, T ], ω ∈ Ω and every N ∈ N. Blömker and Jentzen [2] obtained the following Theorem.…”

“…Our aim here is to extend the result of [2]. First we discuss the case of colored noise not diagonal with respect to the eigenfunctions of the Laplacian.…”

“…Following the ideas of Blömker & Jentzen [2] for the case of space-time white noise, a numerical method for simulating nonlinear SPDEs with additive noise for the case of colored noise is proposed and analyzed. The main novelty in this article is to estimate the spatial and temporal discretization error in the L ∞ -topology in the case of colored noise.…”

“…Recently, Blömker and Jentzen [2] obtained the spatial discretization error in uniform topology by the spectral Galerkin method for the case of space-time white noise.…”