AI 中文总结
本文针对抛物型随机偏微分方程，通过建立离散空间算子的一致$H^\infty$-演算，证明了全离散格式的随机极大$L^p$-正则性及最优估计，并推广至空间半离散情形。
AI 中文摘要
本文建立了抛物型随机偏微分方程数值逼近的全离散随机极大$L^p$-正则性及相应的尖锐极大估计。我们考虑在$\mathbb{R}^3$中光滑、有界、凸区域上具有Dirichlet边界条件的一般二阶椭圆算子$A=-\nabla \cdot a\nabla +b\cdot \nabla +c$的空间有限元离散$A_h$，并耦合包括有理逼近和指数Euler方法在内的广泛时间格式。为获得这些最优离散正则性结果，我们建立了离散空间算子$A_h$的一致于网格尺寸$h$的有界$H^\infty$-演算。作为直接副产品，我们还建立了相应空间半离散格式的空间离散随机极大正则性。
英文摘要
This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations.
