An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation (Q6624855)

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 align with theoretical expectations, thus affirming the reliability of the proposed method.\N\NThe significance of this paper lies in its contribution to numerical methods for solving partial differential equations, especially within the framework of finite element methods for complex, multidimensional systems. By providing a highly accurate and computationally efficient ADI scheme for the linear Schrödinger equation, this study makes a valuable addition to the field of numerical analysis and computational physics, presenting a viable solution for researchers and practitioners working with similar systems in applied mathematics and engineering.