Galerkin Approximations for the Stochastic Burgers Equation

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 the estimation of the difference of the finite-dimensional Galerkin approximations and of the solution of the infinite-dimensional SPDE uniformly in space, i.e., in the $L^\infty$-topology, instead of the usual Hilbert space estimates in the $L^2$-topology, that were shown before.