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STABILIZED, MULTISCALE, AND MULTIPHYSICS METHODS IN FLUID MECHANICS

Stabilized Finite Element Methods for the Schrödinger Wave Equation

[+] Author and Article Information
Raguraman Kannan

Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607

Arif Masud1

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801amasud@uiuc.edu

1

Corresponding author.

J. Appl. Mech 76(2), 021203 (Jan 14, 2009) (7 pages) doi:10.1115/1.3059564 History: Received November 27, 2007; Revised July 10, 2008; Published January 14, 2009

This paper presents two stabilized formulations for the Schrödinger wave equation. First formulation is based on the Galerkin/least-squares (GLS) method, and it sets the stage for exploring variational multiscale ideas for developing the second stabilized formulation. These formulations provide improved accuracy on cruder meshes as compared with the standard Galerkin formulation. Based on the proposed formulations a family of tetrahedral and hexahedral elements is developed. Numerical convergence studies are presented to demonstrate the accuracy and convergence properties of the two methods for a model electronic potential for which analytical results are available.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A family of 3D linear and quadratic elements

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Figure 2

Convergence rates for eigenvalues using linear brick elements (GLS)

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Figure 3

Convergence rates for eigenvalues using linear tetrahedral elements (GLS)

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Figure 4

Convergence rates for eigenvalues using quadratic brick elements (GLS)

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Figure 5

Convergence rates for eigenvalues using quadratic tetrahedral elements (GLS)

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Figure 6

Convergence rates for eigenvalues using linear brick elements (HVM)

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Figure 7

Convergence rates for eigenvalues using linear tetrahedral elements (HVM)

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Figure 8

Convergence rates for eigenvalues using quadratic brick elements (HVM)

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Figure 9

Convergence rate for eigenvalues using quadratic tetrahedral elements (HVM)

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Energy band diagram for the GLS formulation

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Energy band diagram for the HVM formulation

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Figure 12

Convergence rates for eigenvalues using linear brick elements (GLS)

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Figure 13

Convergence rates for eigenvalues using linear tetrahedral elements (GLS)

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Figure 14

Convergence rates for eigenvalues using quadratic brick elements (GLS)

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Figure 15

Convergence rates for eigenvalues using quadratic tetrahedral elements (GLS)

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Figure 16

Convergence rates for eigenvalues using linear brick elements (HVM)

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Figure 17

Convergence rates for eigenvalues using linear tetrahedral elements (HVM)

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Figure 18

Convergence rates for eigenvalues using quadratic brick elements (HVM)

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Figure 19

Convergence rates for eigenvalues using quadratic tetrahedral elements (HVM)

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