We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn--Hilliard equation setting for the first time.
翻译:我们证明了一种针对带有加性噪声的随机Cahn-Hilliard方程的全离散方案的弱收敛性,其中在空间中使用了谱Galerkin方法,在时间中使用了反向欧拉方法。与Allen-Cahn类型的随机偏微分方程相比,由于非线性项前面存在无界算子,因此这里的误差分析要复杂得多。为了解决这些问题,我们采用了一种新颖和直接的方法,它不依赖于Kolmogorov方程,而是依赖于Malliavin演算法的分部积分公式。据我们所知,这是首次在随机Cahn-Hilliard方程的设置中揭示了弱收敛性的速率。