Finite Element Approximations of Stochastic Linear Schrödinger equation driven by additive Wiener noise

TL;DR

This work analyzes spatial semi-discrete finite element approximations of the stochastic linear Schrödinger equation with additive Wiener noise on bounded convex polygonal domains. It establishes strong convergence rates by first addressing the deterministic problem (nonhomogeneous and homogeneous) and then the stochastic problem, using Ritz and $L^2$ projections along with semigroup techniques. The key contributions are explicit $L^2(\Omega;\dot{H}^0)$ error bounds of order $h^{\beta/2}$ (for $\beta\in[0,4]$) and a rigorous treatment of noise via covariance operators $Q_i$, validated by numerical experiments that illustrate convergence behavior. The results provide a systematic finite element framework for SPDEs of Schrödinger type with additive noise and pave the way for extensions to nonlinear or multiplicative-noise settings and fully discrete schemes.

Abstract

In this article, we have analyzed semi-discrete finite element approximations of the Stochastic linear Schrödinger equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and derive an error estimate with respect to the discretization parameter of the finite element approximation. Numerical experiments have also been performed to support theoretical bounds.

Figures (2)

• Figure 1: The order of strong convergence in $L^2$-norm for deterministic problem
• Figure 2: The order of strong convergence in $L^2$-norm for stochastic problem

Theorems & Definitions (14)

• Definition 2.1: Weak solution of system \ref{['system3']}
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2: prato, Weak Solution
• proof : Proof of Theorem \ref{['est_appr_1']}
• proof : Proof of Theorem \ref{['det_err1']}
• proof : Proof of Theorem \ref{['det_err2']}