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

This thesis is primarily concerned with various issues regarding finite element approximation of the time-dependent Ginzburg-Landau equations. <p />The time-dependent Ginzburg-Landau equations is a macroscopic, phenomenological model of superconductivity, consisting of a system of nonlinear, parabolic partial differential equations. <p />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. <p />A posteriori error estimates are proven in L^inf(L<font size="-1">²</font>), L<font size="-1">²</font>(L<font size="-1">²</font>) and L<font size="-1">²</font>(H<font size="-1">¹</font>)-norms. In particular, error bounds are obtained for certain quantities of physical interest, such as the magnetic field and the magnetization. <p />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. <p />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. <p />Local a priori error estimates of temporal residual quantities are proven. <p />The sharpness of the a posteriori error bounds is investigated numerically in the context of a linear model problem