Discrete maximal regularity of an implicit Euler--Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

TL;DR

This paper establishes a discrete maximal regularity result for an implicit Euler--Maruyama scheme with non-uniform time steps applied to stochastic PDEs, using spectral spatial discretization and Wiener process truncation.

Contribution

It introduces a novel discrete maximal regularity framework for the Euler--Maruyama scheme with non-uniform steps in stochastic PDEs, mirroring continuous regularity properties.

Findings

Proves a discrete maximal $L^2$-regularity for the scheme.

Shows the regularity of the discretized stochastic convolution.

Demonstrates the scheme's effectiveness with spectral spatial discretization.

Abstract

An implicit Euler--Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.