An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation
type: Item
instance of: scholarly article
title: An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation
author: Morrakot Khebchareon
author: Amiya K. Pani
author: Graeme Fairweather
author: Ryan I. Fernandes
published in: Numerical Algorithms
publication date: 2024-10-28T00:00:00Z
review text: The manuscript addresses a finite element method (FEM) for solving the linear Schrödinger equation. The paper presents a novel alternating direction implicit (ADI) Galerkin finite element scheme specifically applied to a Schrödinger-type system over a unit square, where the time integration utilises an extrapolated Crank-Nicolson ADI approach, while the spatial discretisation is managed through a finite element Galerkin method.\N\NThe study addresses the need for efficient and accurate computational methods for linear Schrödinger-type systems, which have a wide range of applications in quantum mechanics, plasma physics, and seismology. Specifically, this study focuses on an initial-boundary value problem on a bounded domain, which is both computationally challenging and significant due to the complex oscillatory nature of solutions in such systems. The ADI finite element Galerkin (FEG) method proposed in this study provides a robust computational approach that enhances stability and precision by transforming the original two-dimensional problem into simpler one-dimensional problems at each time step, which can be solved independently. This dimensional reduction substantially reduces the computational cost and complexity associated with solving the Schrödinger equation on a multidimensional domain.\N\NIn the paper, the authors provide rigorous mathematical proof for the existence and uniqueness of the approximate solution derived by their method, demonstrating that the scheme achieves optimal-order convergence in space and is second-order accurate in time. Their analysis employs various norm-based error metrics, including \(L^2\), \(H^1\), and \(L^\infty\) norms, to establish the accuracy and robustness of the ADI FEG method. This theoretical framework is supported by numerical experiments that illustrate the optimal-order convergence of the method and validate its effectiveness in capturing the oscillatory behaviour of the Schrödinger equation solutions. The results…
reviewed by: Denys Dutykh
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 7932421
zbMATH Keywords: linear Schrödinger equation
zbMATH Keywords: Schrödinger-type systems
zbMATH Keywords: finite element Galerkin method
zbMATH Keywords: extrapolated Crank-Nicolson method
zbMATH Keywords: alternating direction implicit method
MaRDI profile type: Publication
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DOI: 10.1007/S11075-023-01740-5
description: scientific article; zbMATH DE number 7932421