This courses gives an introduction to the co-called Ginzburg-Landau equation, which is a common model to describe superconductors (i.e. materials that allow to conduct electricity without any electrical resistance). We will study the equation both from an analytical and a numerical perspective. Special attention is given to the Ginzburg-Landau parameter which is a material parameter that triggers the appearance of quantized vortices in the solution. Regarding the analysis of the equation, we are concerned with the existence and uniqueness of solutions, as well as their stability in different Lebesgue and Sobolev norms. Regarding the numerics, we study the approximation of solutions in finite element spaces including error estimates that are explicit with respect to the Ginzburg-Landau parameter. Since the Ginzburg-Landau equation is a nonlinear partial differential equation, we will also investigate suitable iterative solvers for the arising nonlinear systems. Finally, we will also deal with aspects regarding the practical implementation of the numerical methods.
Prerequisites:
Analysis I-III and knowledge about numerical methods for differential equations (for example, "Introduction to Numerics“ + "Numerics for ordinary differential equations"). Knowledge about the theory of partial differential equations and functional analysis is helpful, but not crucially necessary.
Literature:
Will be announced in the lecture.
- Kursleiter/in: Patrick Henning