# Strong Convergence of the Linear Implicit Euler Method for the Finite Element Discretization of Semilinear non-Autonomous SPDEs Driven by Multiplicative or Additive Noise

## TL;DR

This paper establishes strong convergence rates for a finite element and linear implicit Euler discretization of non-autonomous semilinear SPDEs with multiplicative or additive noise, extending understanding beyond autonomous cases.

## Contribution

It provides the first strong convergence analysis for fully discrete schemes applied to non-autonomous SPDEs with detailed dependence on initial data and noise regularity.

## Key findings

- Convergence order for multiplicative noise: O(h^{2-ε} + Δt^{1/2})
- Convergence order for additive noise: O(h^{2-ε} + Δt^{1-ε})
- Numerical experiments confirm theoretical convergence rates.

## Abstract

This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs while modeling real world phenomena. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. The non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semi group and provide the strong convergence result of the fully discrete scheme toward the exact solution in the root-mean-square $L^2$ norm. The results indicate how the converge order depends on the regularity of the initial solution and the noise. In particular, for multiplicative trace class noise we achieve convergence order $\mathcal{O}(h^{2-\epsilon}+\Delta t^{1/2})$ and for additive noise with trace class, we achieve convergence order $\mathcal{O}(h^{2-\epsilon}+\Delta t^{1-\epsilon})$, for an arbitrarily small $\epsilon>0$. Numerical experiments to sustain our theoretical results are provided.