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

TL;DR

The paper addresses the numerical analysis of a linear stochastic heat equation on Banach spaces by discretizing the spatial variable with FEM. It proves that the discrete negative Laplacian $\mathcal{A}_h$ has a uniformly bounded $H^\infty$-calculus with respect to the mesh size $h$, enabling $h$-uniform control of imaginary powers and $\mathcal{R}$-bounded resolvent families. Building on this, it establishes discrete stochastic maximal $L^p$-regularity for stochastic convolutions and derives convergence rates for the spatial semidiscretization in general spatial $L^q$ norms, including near-optimal pathwise uniform convergence. The results extend stochastic maximal regularity methods from Hilbert spaces to general Banach spaces and provide a robust toolset for nonlinear SPDE analysis based on FE discretizations. Overall, the work lays a rigorous foundation for stable and accurate SPDE simulations in Banach-space settings.

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.

Figures (1)

• Figure 4.1: $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$.

Theorems & Definitions (22)

• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Theorem 3.3
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• proof
• Lemma 4.4
• proof