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

TL;DR

This paper proves uniform boundedness of the discrete negative Laplace operator's H-infinity calculus and uses it to establish maximal L^p-regularity and convergence results for a spatial discretization of a stochastic heat equation.

Contribution

It demonstrates the uniform boundedness of the H-infinity calculus for the finite element discretization of the Laplace operator and derives new regularity and convergence estimates for the stochastic heat equation.

Findings

Uniform boundedness of the H-infinity calculus for the discrete Laplacian

Discrete stochastic maximal L^p-regularity estimate established

Nearly optimal pathwise convergence rate in spatial L^q-norms

Abstract

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.