On the well-posedness of the stochastic Allen-Cahn equation in two dimensions

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.Comment: 21 pages, 4 figures; accepted by Journal of Computational Physics (Dec 2011

Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation

We study a class of fully-discrete schemes for the numerical approximation of solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and driven by additive noise. The spatial (resp. temporal) discretization is performed with a spectral Galerkin method (resp. a tamed exponential Euler method). We consider two situations: space-time white noise in dimension $d=1$ and trace-class noise in dimensions $d=1,2,3$. In both situations, we prove weak error estimates, where the weak order of convergence is twice the strong order of convergence with respect to the spatial and temporal discretization parameters. To prove these results, we show appropriate regularity estimates for solutions of the Kolmogorov equation associated with the stochastic Cahn--Hilliard equation, which have not been established previously and may be of interest in other contexts

Strong convergence rates of an explicit scheme for stochastic Cahn-Hilliard equation with additive noise

In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn-Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn-Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.Comment: 24 pages, 3 figure

Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.Comment: 25 page