Stabilized Finite Element Methods for the Schrdinger Wave Equation

This paper presents two stabilized formulations for the Schrodinger 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. DOI: 10.1115/1.3059564

[1]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[2]  A. Masud,et al.  A variational multiscale method for inelasticity: Application to superelasticity in shape memory alloys , 2006 .

[3]  Lawrence A. Bergman,et al.  Application of multi-scale finite element methods to the solution of the Fokker–Planck equation , 2005 .

[4]  J. Pask,et al.  Finite-element methods in electronic-structure theory , 2001 .

[5]  Thomas J. R. Hughes,et al.  A stabilized mixed finite element method for Darcy flow , 2002 .

[6]  H. Chermette.  Density functional theory , 1998 .

[7]  Y. Saad,et al.  Finite-difference-pseudopotential method: Electronic structure calculations without a basis. , 1994, Physical review letters.

[8]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[9]  C. Y. Fong,et al.  Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach , 1999, cond-mat/9903313.

[10]  Dong Qian,et al.  Mechanics of carbon nanotubes , 2002 .

[11]  L. Franca,et al.  A hierarchical multiscale framework for problems with multiscale source terms , 2008 .

[12]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[13]  Wing Kam Liu,et al.  Moving least-square basis for band-structure calculations of natural and artificial crystals , 2007 .

[14]  Kumar Vemaganti.  Local error estimation for dimensionally reduced models of elliptic boundary value problems , 2003 .

[15]  Wu,et al.  Higher-order finite-difference pseudopotential method: An application to diatomic molecules. , 1994, Physical review. B, Condensed matter.

[16]  Harold S. Park,et al.  An introduction to computational nanomechanics and materials , 2004 .

[17]  H. Johnson,et al.  The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes , 2004 .

[18]  Richard M. Martin.  Electronic Structure: Frontmatter , 2004 .

[19]  Arif Masud,et al.  A multiscale finite element method for the incompressible Navier-Stokes equations , 2006 .

[20]  R. Pierret,et al.  Advanced semiconductor fundamentals , 1987 .

[21]  R. Martin,et al.  Electronic Structure: Basic Theory and Practical Methods , 2004 .