Discrete stochastic maximal $ L^p $-regularity and convergence of a spatial semidiscretization for a linear stochastic heat equation

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.