论文标题
随机Cahn-Hilliard方程的全差异方案的弱误差估计
Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation
论文作者
论文摘要
我们研究了一类全差异方案，用于具有立方非线性的随机cahn--hilliard方程溶液的数值近似，并由添加噪声驱动。空间（分别时间）离散化是通过光谱盖金方法（分别驯服的指数Euler方法）进行的。我们考虑两种情况：尺寸的时空白噪声$ d = 1 $和尺寸的微量级噪声$ d = 1,2,3 $。在这两种情况下，我们都证明了弱误差估计值，其中弱收敛顺序是相对于空间和时间离散参数的强大收敛顺序的两倍。为了证明这些结果，我们显示了与随机cahn--hilliard方程相关的Kolmogorov方程解决方案解决方案的适当规律性估计，这些方程前尚未建立，并且在其他情况下可能会引起人们的关注。
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.
