On the Finite Element Method for the Time-Dependent Ginzburg-Landau Equations

Doktorsavhandling, 2001

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(L2), L2(L2) and L2(H1)-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.