Manchester Applied Mathematics and Numerical Analysis Seminars Autumn 2000/2001

November 8, 2000, 2.30 pm

Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Analysis of vortex density models in superconductivity

Dr. Vanessa Styles, School of Mathematical Sciences, University of Sussex

The vortex density model of superconducting vortices, derived by S.J.Chapman in 1995, comprises of a coupling between a first order conservation law and an elliptic partial differential equation involving two variables omega and H that respectively denote the vortex density, commonly known as the 'vorticity', and the avarage magnetic field of the superconducting sample. We study one and two-dimensional forms of this model, and prove the existence of a weak solution and a steady state solution, which takes the form of a free boundary problem. For the one-dimensional model we prove that the weak solution is unique and that it satifies an entropy inequality. We derive a finite-volume discretization of the one and two-dimensional models. For the one-dimensional model we prove that in the limit as the mesh size and the time step tend to zero, our numerical approximation converges to the unique weak solution of the model. Lastly we introduce some generalisations of the model which deal with nucleation of vorticity at the boundary and flux pinning.

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk) or the seminar secretary (Tel. 0161 275 5800).