Top

Journal of Scientific Computing

Published in:

08-04-2015

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

Authors: Bo Dong, Chi-Wang Shu, Wei Wang

Published in: Journal of Scientific Computing | Issue 1/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

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.

previous article Petviashvilli’s Method for the Dirichlet Problem

next article An HDG Method for Convection Diffusion Equation

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Aarnes, J., Heimsund, B.-O.: Multiscale discontinuous Galerkin methods for elliptic problems with multiple scales. In: Multiscale Methods in Science and Engineering, Lecture Notes in Computer Science Engineering, vol. 44, pp. 1–20. Springer, Berlin (2005)

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetCrossRefMATH

Arnold, A., Ben Abdallah, N., Negulescu, C.: WKB-based schemes for the oscillatory 1D Schrödinger equation in the semiclassical limit. SIAM J. Numer. Anal. 49, 1436–1460 (2011)MathSciNetCrossRefMATH

Ben Abdallah, N., Mouis, M., Negulescu, C.: An accelerated algorithm for 2D simulations of the quantum ballistic transport in nanoscale MOSFETs. J. Comput. Phys. 225, 74–99 (2007)MathSciNetCrossRefMATH

Ben Abdallah, N., Pinaud, O.: Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation. J. Comput. Phys. 213, 288–310 (2006)MathSciNetCrossRefMATH

Bohm, D.: Quantum Theory. Dover, New York (1989)

Buffa, A., Monk, P.: Error estimates for the ultra weak variational formulation of the helmholtz equation. ESAIM M2AN Math Model. Numer. Anal. 42, 925–940 (2008)MathSciNetCrossRefMATH

Cockburn, B., Dong, B.: An analysis of the minimal dissipation local discontinuous Galerkin method for convection–diffusion problems. J. Sci. Comput. 32, 233–262 (2007)MathSciNetCrossRefMATH

Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)MathSciNetCrossRefMATH

10.

Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47, 2872–2896 (2009)MathSciNetCrossRefMATH

11.

Gabard, G.: Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225, 1961–1984 (2007)MathSciNetCrossRefMATH

12.

Gittelson, C., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM M2AN Math Model. Numer. Anal. 43, 297–331 (2009)MathSciNetCrossRefMATH

13.

Lent, C.S., Kirkner, D.J.: The quantum transmitting boundary method. J. Appl. Phys. 67, 6353–6359 (1990)CrossRef

14.

Negulescu, C.: Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation. Numer. Math. 108, 625–652 (2008)MathSciNetCrossRefMATH

15.

Negulescu, C., Ben Abdallah, N., Polizzi, E., Mouis, M.: Simulation schemes in 2D nanoscale MOSFETs: a WKB based method. J. Comput. Electron. 3, 397–400 (2004)CrossRef

16.

Polizzi, E., Ben Abdallah, N.: Subband decomposition approach for the simulation of quantum electron transport in nanostructures. J. Comput. Phys. 202, 150–180 (2005)MathSciNetCrossRefMATH

17.

Wang, W., Guzmán, J., Shu, C.-W.: The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients. Int. J. Numer. Anal. Model 8, 28–47 (2011)MathSciNetMATH

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)MathSciNetCrossRefMATH

19.

Yuan, L., Shu, C.-W.: Discontinuous Galerkin method based on non-polynomial approximation spaces. J. Comput. Phys. 218, 295–323 (2006)MathSciNetCrossRefMATH

20.

Yuan, L., Shu, C.-W.: Discontinuous Galerkin method for a class of elliptic multi-scale problems. Int. J. Numer. Methods Fluids 56, 1017–1032 (2008)MathSciNetCrossRefMATH

21.

Zhang, Y., Wang, W., Guzmán, J., Shu, C.-W.: Multi-scale discontinuous Galerkin method for solving elliptic problems with curvilinear unidirectional rough coefficients. J. Sci. Comput. 61, 42–60 (2014)MathSciNetCrossRefMATH

Title: A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation
Authors: Bo Dong
Chi-Wang Shu
Wei Wang
Publication date: 08-04-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0022-7

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2016

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control

A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

Premium Partner