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 $H^{\mathrm{\infty }}$-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 $H^{\mathrm{\infty }}$-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.