On the Finite Element Method for the Time-Dependent Ginzburg-Landau Equations
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Johan Ivarsson
Department of Mathematics
Chalmers University of Technology
Göteborg University

Supervisor: Stig Larsson

Abstract:

This thesis is primarily concerned with various issues regarding
finite element approximation of the time-dependent Ginzburg-Landau
equations.

The time-dependent Ginzburg-Landau equations is a macroscopic,
phenomenological model of superconductivity, consisting of a system of
nonlinear, parabolic partial differential equations.

These equations are discretized under the Lorentz gauge using standard
continuous piecewise linear discretization in space and the
discontinuous Galerkin method in time, with arbitrary polynomial
order.

A posteriori error estimates are proven in L^inf(L^2), L^2(L^2) and
L^2(H^1)-norms.  In particular, error bounds are obtained for certain
quantities of physical interest, such as the magnetic field and the
magnetization.

The a posteriori error estimates are expressed in terms of certain
residual quantities and the regularity properties of a related
linearized dual problem.  Several results concerning the these
regularity properties are shown.  An averaging interpolation operator
which preserves certain boundary conditions and requires low
regularity is constructed.

Existence and uniqueness of solutions the discrete nonlinear equations
are shown.  Algorithms for solving these equations are proposed and
analyzed.  Adaptivity based on the a posteriori error estimates are
discussed.  Numerous numerical studies are presented, in particular,
investigating the behaviour of the residual quantities and the
adaptivity.

Local a priori error estimates of temporal residual quantities are
proven.

The sharpness of the a posteriori error bounds is investigated
numerically in the context of a linear model problem.

Keywords: discontinuous Galerkin method, a posteriori error estimate,
adaptive finite element method, duality, residual, Ginzburg-Landau
equations, superconductivity