Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise

TL;DR

This paper develops a finite element and Magnus-type integrator scheme for non-autonomous semilinear parabolic SPDEs with multiplicative noise, proving strong convergence and analyzing how regularity affects convergence rates.

Contribution

It introduces a novel numerical scheme combining finite element and Magnus-type integrator methods for non-autonomous SPDEs with multiplicative noise, with proven convergence analysis.

Findings

Achieves convergence order of O(h^2 + Δt^{1/2}) for trace class noise.

Provides a strong convergence proof in the root-mean-square L^2 norm.

Numerical simulations confirm theoretical convergence rates.

Abstract

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+\Delta t^{1/2}\right)$. Numerical simulations to illustrate our theoretical finding are provided.