Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise

Abstract

Abstract We consider the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild space–time noise posed on a bounded domain of $\mathbb{R}^2$. The positive parameter $\varepsilon $ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin., 15, 407–438). The noise, although smooth, becomes white on the sharp interface limit $\varepsilon \rightarrow 0^+$. We construct a nonlinear discontinous Galerkin scheme with space–time finite elements of general type that are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer’s Theorem. We first derive abstract error estimates and then, for the case of piecewise polynomial finite elements, we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\varepsilon $. Finally, we present a linear approximation of the nonlinear scheme, for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in $2+1$-dimensional subdomains for a nonlinear parabolic problem. In addition this problem involves noise. These types of schemes avoid any Runge–Kutta-type discretization for the evolutionary variable, and seem to be very effective when applied to equations of such a difficulty.