Fully discrete stochastic maximal regularity and H∞-calculus for second-order elliptic operators
Abstract
This paper establishes the fully discrete stochastic maximal Lp-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization Ah of a general second-order elliptic operator A=−∇ · a∇ +b· ∇ +c with Dirichlet boundary conditions on a smooth, bounded, convex domain in ℝ3, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded H∞-calculus for the discrete spatial operator Ah, uniformly in the mesh size h. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen--Cahn equation with variable diffusion coefficients and multiplicative noise.
Related
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BibTeX
@misc{evangelopoulosntemiris2026fully,
title = {{Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators}},
author = {Foivos Evangelopoulos-Ntemiris},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07436},
eprint = {2609.07436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07436},
}
RIS
TY - UNPB
TI - Fully discrete stochastic maximal regularity and H^∞-calculus for second-order elliptic operators
AU - Foivos Evangelopoulos-Ntemiris
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.07436
UR - https://arxiv.org/abs/2609.07436
AB - This paper establishes the fully discrete stochastic maximal L^p-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization A_h of a general second-order elliptic operator A=−∇ · a∇ +b· ∇ +c with Dirichlet boundary conditions on a smooth, bounded, convex domain in ℝ^3, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded H^∞-calculus for the discrete spatial operator A_h, uniformly in the mesh size h. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen--Cahn equation with variable diffusion coefficients and multiplicative noise.
ER -