{"title":"A space–time DG method for the Schrödinger equation with variable potential","authors":"Sergio Gómez, Andrea Moiola","doi":"10.1007/s10444-024-10108-9","DOIUrl":null,"url":null,"abstract":"<div><p>We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal <i>h</i>-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 2","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10108-9.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10444-024-10108-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.
我们提出了一种具有可变势能的线性薛定谔方程的时空超弱非连续伽勒金离散化方法。对于非常一般的离散空间，所提出的方法在网格相关规范中具有良好的假设性和准最优性。当测试和试验空间选择为片断多项式空间或新的准特列夫兹多项式空间时，可得出该方法的最佳 h 收敛误差估计值。后者允许大幅减少自由度数量，并允许片滑势垒。几个数值实验验证了所提方法的准确性和优势。
期刊介绍：
Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis.
This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.
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