Adaptive Galerkin Methods with Error Control for a Dynamical Ginzburg--Landau Model in Superconductivity

The time-dependent Ginzburg--Landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations. It is discretized semi-implicitly in time and in space via continuous piecewise linear finite elements. A posteriori error estimates are derived for the L^∞ L² norm by studying a dual problem of the linearization of the original system, other than the dual of error equations. Numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method.