Magnus-type integrator for the finite element discretization of semilinear parabolic non-autonomous SPDEs driven by additive noise

TL;DR

This paper develops and analyzes a Magnus-type integrator combined with finite element spatial discretization for non-autonomous semilinear parabolic SPDEs with additive noise, achieving optimal convergence rates.

Contribution

It introduces a novel numerical scheme for non-autonomous SPDEs and proves its strong convergence with optimal order, extending results beyond autonomous cases.

Findings

Achieves strong convergence with order > 1/2 in time.

Optimal spatial convergence order of approximately h^2.

Numerical simulations confirm theoretical convergence rates.

Abstract

In this paper, we investigate a numerical approximation of a general second order semilinear parabolic non-autonomous stochastic partial differential equation (SPDE) driven by additive noise. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature while the non-autonomous case is not yet well understood. We discretize the non-autonomous SPDE in space by the finite element method and in time by the Magnus-type integrator. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. Appropriate assumptions on the drift term and the noise allow to achieve optimal convergence order in time greater than $1/2$, without any logarithmic reduction of convergence order in time. In particular, for trace class noise, we achieve optimal convergence orders $\mathcal{O}\left(h^{2-\epsilon}+\Delta t\right)$, where $\epsilon$ is a positive number small enough. Numerical simulations are provided to illustrate our theoretical results.