Convergence of the spectral Galerkin method for the stochastic reaction–diffusion–advection equation

Abstract

We study the convergence of the spectral Galerkin method in solving the stochastic reaction–diffusion–advection equation under different Lipschitz conditions of the reaction function f. When f is globally (locally) Lipschitz continuous, we prove that the spectral Galerkin approximation strongly (weakly) converges to the mild solution of the stochastic reaction–diffusion–advection equation, and the rate of convergence in Hr-norm is (12−r)−, for any r∈[0,12) (r∈(12−12d,12)). The convergence analysis in the local Lipschitz case is challenging, especially in the presence of an advection term. We propose a new approach based on the truncation techniques, which can be easily applied to study other stochastic partial differential equations. Numerical simulations are also provided to study the convergence of Galerkin approximations.