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We consider the ε-dependent stochastic Allen-Cahn equation with mild spacetime noise posed on a bounded domain of R. The positive parameter ε 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 in [15]. The noise although smooth becomes white on the sharp interface limit ε → 0. We construct a nonlinear dG scheme with space-time finite elements of general type which 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 piece-wise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter ε. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piece-wise 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 type 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.

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