A posteriori error estimation and space-time adaptivity for a linear stochastic PDE with additive noise

Abstract

Abstract We present a strong residual-based a posteriori error estimate for a finite element-based space-time discretization of the linear stochastic convected heat equation with additive noise. This error estimate is used for an adaptive algorithm that automatically selects deterministic mesh parameters in space and time. For every $n \geq 0$, we find a new time-step $\tau _n$, a new spatial mesh ${\mathcal M}_{n}$ terminating within finitely many iterations and a finite element value approximation $Y^n_h$ on this spatial mesh, which then approximates strongly the solution of the stochastic partial differential equation (SPDE) within a prescribed tolerance.