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Convergence Analysis of the Localized Orthogonal Decomposition Method for the Semiclassical Schrödinger Equations with Multiscale Potentials

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Abstract

We provide a convergence analysis of the localized orthogonal decomposition (LOD) method for Schrödinger equations with general multiscale potentials in the semiclassical regime. We focus on the influence of the semiclassical parameter and the multiscale structure of the potential on the numerical scheme. We construct localized multiscale basis functions by solving a constrained energy minimization problem in the framework of the LOD method. We show that localized multiscale basis functions with a smaller support can be constructed as the semiclassical parameter approaches zero. We obtain the first-order convergence in the energy norm and second-order convergence in the \(L^2\) norm for the LOD method and super convergence rates can be obtained if the solution possesses sufficiently high regularity. We analyse the temporal and spatial regularity of the solution and find that the spatial derivatives are more oscillatory than the time derivatives in the presence of a multiscale potential. By combining the regularity analysis, we are able to derive the dependence of the error estimates on the small parameters of the Schrödinger equation. Moreover, we find that the LOD method outperforms the finite element method in the presence of a multiscale potential due to the super convergence for high-regularity solutions and more relaxed dependence of the error estimates on the small parameters for low-regularity solutions. Finally, we present numerical results to demonstrate the accuracy and robustness of the proposed method.

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The data generated and analysed during the current study are available from the corresponding author on reasonable request.

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Custom code was written in MATLAB.

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Acknowledgements

The research of Z. Zhang is supported by Hong Kong RGC grants (Projects 17300817, 17300318, and 17307921), National Natural Science Foundation of China (Project 12171406), Seed Funding Programme for Basic Research (HKU), and a seed funding from the HKU-TCL Joint Research Center for Artificial Intelligence. The computations were performed using research computing facilities offered by Information Technology Services, the University of Hong Kong.

Funding

The funding was provided by Hong Kong RGC grants (Projects 17300817, 17300318, and 17307921), National Natural Science Foundation of China (Project 12171406), Seed Funding Programme for Basic Research (HKU), and a seed funding from the HKU-TCL Joint Research Center for Artificial Intelligence. The computations were performed using research computing facilities offered by Information Technology Services, the University of Hong Kong.

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  1. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, People’s Republic of China

    Zhizhang Wu & Zhiwen Zhang

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  1. Zhizhang Wu
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Wu, Z., Zhang, Z. Convergence Analysis of the Localized Orthogonal Decomposition Method for the Semiclassical Schrödinger Equations with Multiscale Potentials. J Sci Comput 93, 73 (2022). https://doi.org/10.1007/s10915-022-02038-9

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