This study investigates the boundedness of the \( H^\infty \)-calculus for the discrete negative Laplace operator, subject to homogeneous Dirichlet boundary conditions. The discrete negative Laplace operator is implemented using the finite element method, and we establish that its \(H^\infty\)-calculus is uniformly bounded with respect to the spatial mesh size. Using this finding, we derive a discrete stochastic maximal \(L^p\)-regularity estimate for a spatial semidiscretization of a linear stochastic heat equation. Furthermore, we provide a nearly optimal pathwise uniform convergence estimate for this spatial semidiscretization within the framework of general spatial \(L^q\)-norms.

翻译：本研究探讨了在齐次 Dirichlet 边界条件下离散负 Laplace 算子的 \( H^\infty \) 演算的有界性。该离散负 Laplace 算子采用有限元方法实现，我们证明了其 \(H^\infty\)-演算关于空间网格尺寸是一致有界的。利用这一结果，我们针对线性随机热方程的空间半离散格式推导出一个离散随机极大 \(L^p\) 正则性估计。此外，我们在一般空间 \(L^q\) 范数的框架下，为该空间半离散格式提供了一个近乎最优的路径一致收敛估计。