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Full Discretization of Stochastic Semilinear Schrödinger equation driven by multiplicative Wiener noise

Suprio Bhar, Mrinmay Biswas, Mangala Prasad

TL;DR

This work investigates the full discretization of the stochastic semilinear Schrödinger equation with multiplicative noise on bounded convex polygonal domains. It combines finite element spatial discretization with a stochastic trigonometric method for time integration to obtain strong mean-square convergence bounds in both space and time for linear and semilinear cases. The authors provide rigorous proofs of convergence and support the theory with numerical experiments that confirm the predicted rates. This advances the numerical analysis of stochastic Schrödinger equations by delivering a new full discretization framework for the semilinear, multiplicative-noise setting and demonstrating its reliability in practice.

Abstract

In this article, we have analyzed the full discretization of the Stochastic semilinear Schrödinger equation in a bounded convex polygonal domain driven by multiplicative Wiener noise. We use the finite element method for spatial discretization and the stochastic trigonometric method for time discretization and derive a strong convergence rate with respect to both parameters (temporal and spatial). Numerical experiments have also been performed to support theoretical bounds.

Full Discretization of Stochastic Semilinear Schrödinger equation driven by multiplicative Wiener noise

TL;DR

This work investigates the full discretization of the stochastic semilinear Schrödinger equation with multiplicative noise on bounded convex polygonal domains. It combines finite element spatial discretization with a stochastic trigonometric method for time integration to obtain strong mean-square convergence bounds in both space and time for linear and semilinear cases. The authors provide rigorous proofs of convergence and support the theory with numerical experiments that confirm the predicted rates. This advances the numerical analysis of stochastic Schrödinger equations by delivering a new full discretization framework for the semilinear, multiplicative-noise setting and demonstrating its reliability in practice.

Abstract

In this article, we have analyzed the full discretization of the Stochastic semilinear Schrödinger equation in a bounded convex polygonal domain driven by multiplicative Wiener noise. We use the finite element method for spatial discretization and the stochastic trigonometric method for time discretization and derive a strong convergence rate with respect to both parameters (temporal and spatial). Numerical experiments have also been performed to support theoretical bounds.

Paper Structure

This paper contains 12 sections, 7 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Setup and Main results: Full Discretization for the Linear Case
  3. Main results: Full Discretization for the Semilinear Case
  4. Organization of the paper
  5. Proofs of Main results
  6. Mean convergence analysis of linear equation \ref{['eqn1.2']}
  7. Mean convergence analysis of semilinear equation \ref{['eqn1.1']}
  8. Numerical Experiments
  9. Numerical Example
  10. Notations and Definitions
  11. Abstract framework and regularity
  12. Finite element approximations

Key Result

Theorem 1.1

Let $\theta \in [0,2]$, the covariance operators $Q_i,\,i=1,2$ satisfy $X_0=(u_{0,1},u_{0,2})^\text{T}$ and $X_{0,h}=(u_{h,0,1},u_{h,0,2})^\text{T}=(\mathcal{P}_h u_{0,1},\mathcal{P}_h u_{0,2})^\text{T}$. Let $X=(u_1,u_2)^\text{T}$ and $X_h=(u_{h,1},u_{h,2})^\text{T}$ be given by $Slinear \text{ and } Dslinear$ respectively. Then, the following estimates hold for $t \g where $C_t$ is an increasin

Figures (2)

  • Figure 1: The rate of strong convergence with respect to the space discretized parameter h.
  • Figure 2: The rate of strong convergence with respect to the time discretized parameter k.

Theorems & Definitions (14)

  • Theorem 1.1: Theorem 1.4 in finite
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.1
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm2']}
  • ...and 4 more