Error estimates for a finite-difference approximation of a mean field model of superconducting vortices in one-dimension

2001 ◽  
Vol 21 (3) ◽  
pp. 667-701 ◽  
Author(s):  
V. Styles
Keyword(s):  
Finite Difference ◽  
Error Estimates ◽  
Field Model ◽  
Mean Field ◽  
Finite Difference Approximation ◽  
One Dimension ◽  
Difference Approximation ◽  
Mean Field Model ◽  
Superconducting Vortices

A mean-field model of superconducting vortices

1996 ◽  
Vol 7 (2) ◽  
pp. 97-111 ◽  
Author(s):  
S. J. Chapman ◽  
J. Rubinstein ◽  
M. Schatzman
Keyword(s):  
Steady State ◽  
Field Model ◽  
Local Existence ◽  
Mean Field ◽  
State Problem ◽  
Mean Field Model ◽  
Type Ii Superconductor ◽  
Superconducting Vortices ◽  
Steady State Problem ◽  
Steady State Solutions

A mean-field model for the motion of rectilinear vortices in the mixed state of a type-II superconductor is formulated. Steady-state solutions for some simple geometries are examined, and a local existence result is proved for an arbitrary smooth geometry. Finally, a variational formulation of the steady-state problem is given which shows the solution to be unique.

Analysis of a mean field model of superconducting vortices

1999 ◽  
Vol 10 (4) ◽  
pp. 319-352 ◽  
Author(s):  
R. SCHÄTZLE ◽  
V. STYLES
Keyword(s):  
Weak Solution ◽  
Steady State ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Field Model ◽  
Mean Field ◽  
Two Dimensions ◽  
Mean Field Model ◽  
Superconducting Vortices

We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

2013 ◽  
Vol 13 (5) ◽  
pp. 1357-1388 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Error Estimates ◽  
Difference Method ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

AbstractWe study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

scholarly journals Ill-posedness of the mean-field model of superconducting vortices and a possible regularisation

2000 ◽  
Vol 11 (2) ◽  
pp. 137-152 ◽  
Author(s):  
G. RICHARDSON ◽  
B. STOTH
Keyword(s):  
Stability Analysis ◽  
Linear Stability Analysis ◽  
Field Model ◽  
Mean Field ◽  
Average Density ◽  
Mean Field Model ◽  
Superconducting Vortices ◽  
Vorticity Vector ◽  
Ill Posed ◽  
The Mean

We conjecture that the mean-field model of superconducting vortices given in [10] is ill-posed wherever the electric current j has some component in the same direction as the vorticity vector ω (which gives the average density and direction of the superconducting vortices). The conjecture is illustrated with a linear stability analysis of a certain solution to the model. A regularised model is then proposed, and this is used to demonstrate the instability of force-free steady states in a certain geometry.

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