A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation

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• 作者：Bo Dong ; Chi-Wang Shu ; Wei Wang
• 关键词：Discontinuous Galerkin method ; Multiscale method ; Schrödinger equation
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：66
• 期：1
• 页码：321-345
• 全文大小：702 KB
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  18.Wang, W., Shu, C.-W.: The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode. J. Sci. Comput. 40, 360–374 (2009)MathSciNet CrossRef MATH
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• 作者单位：Bo Dong (1)
  Chi-Wang Shu (2)
  Wei Wang (3)
  
  1. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA
  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA
  3. Department of Mathematics & Statistics, Florida International University, Miami, FL, 33199, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schrödinger equations with open boundary conditions which have highly oscillating solutions. Our method uses a smaller finite element space than the WKB local DG method proposed in Wang and Shu (J Comput Phys 218:295–323, 2006) while achieving the same order of accuracy with no resonance errors. We prove that the DG approximation converges optimally with respect to the mesh size \(h\) in \(L^2\) norm without the typical constraint that \(h\) has to be smaller than the wave length. Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schrödinger equations. Keywords Discontinuous Galerkin method Multiscale method Schrödinger equation