A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation

Abstract

In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $\dot{W}^\varepsilon $ tending to rough as $\varepsilon \rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $\varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $\varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $\varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.