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

Mangala Prasad

TL;DR

This work studies the weak convergence of spatial finite element approximations for a stochastic linear Schrödinger equation with additive Wiener noise. By formulating a general weak-error framework and employing semigroup and operator-norm estimates, it derives a bound showing the weak convergence rate is twice the strong rate under suitable covariance assumptions on the noise and for test functions in $C^2_b(H,\mathbb{R})$. The approach parallels existing weak-convergence analyses for heat and wave equations and yields practical guidance for accurately computing expected functionals of the solution. The results enhance understanding of SPDE discretizations in quantum-like dispersive systems and have implications for reliable numerical simulations of stochastic Schrödinger dynamics.

Abstract

A standard finite element method discretizes the stochastic linear Schrödinger equation driven by additive noise in the spatial variables. The weak convergence of the resulting approximate solution is analyzed, and it is established that the weak convergence rate is twice that of the strong convergence.

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

TL;DR

This work studies the weak convergence of spatial finite element approximations for a stochastic linear Schrödinger equation with additive Wiener noise. By formulating a general weak-error framework and employing semigroup and operator-norm estimates, it derives a bound showing the weak convergence rate is twice the strong rate under suitable covariance assumptions on the noise and for test functions in . The approach parallels existing weak-convergence analyses for heat and wave equations and yields practical guidance for accurately computing expected functionals of the solution. The results enhance understanding of SPDE discretizations in quantum-like dispersive systems and have implications for reliable numerical simulations of stochastic Schrödinger dynamics.

Abstract

A standard finite element method discretizes the stochastic linear Schrödinger equation driven by additive noise in the spatial variables. The weak convergence of the resulting approximate solution is analyzed, and it is established that the weak convergence rate is twice that of the strong convergence.

Paper Structure

This paper contains 7 sections, 2 theorems, 66 equations.

Table of Contents

  1. Introduction and Main Result
  2. Preliminaries
  3. Weak Error Formulation
  4. Proof of Main Result
  5. Notations, Definition of Spaces and Their Norms
  6. Finite Element Approximations
  7. Differentiability

Key Result

Theorem 1.1

Let $X$ and $X_h$ be defined as in eqn1.4 and eqn1.5, respectively. Suppose $\theta \in [0,1],$ and covariance operators $Q_i \text{ for } i=1,\,2$ satisfy Additionally, assume that $\Phi \in C^2_b(H,\,\mathbb{R})$. Then there is a constant $C$, depending on $\Phi,\, X_0,\,Q_i$ but not on $T,\, h$ such that for $T\geq 0$,

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm1.1']}