Johan Ivarsson

A posteriori error analysis of a finite element method for the
time-dependent Ginzburg-Landau equations, 

preprint 1998-24, 
Department of Mathematics, Chalmers University of Technology. 
http://www.math.chalmers.se/~ivarsson/gltnet.ps

Abstract.  We consider the discontinuous Galerkin finite element method
  for the time-dependent Ginzburg-Landau model of superconductivity. The
  model consists of an initial-boundary value problem for a system of
  nonlinear parabolic partial differential equations.  We prove an a
  posteriori error estimate in terms of computable residuals and
  stability factors related to a linearised dual problem.  The a
  posteriori error estimate is suitable for the design of an adaptive
  finite element method.  We also construct an interpolation operator,
  based on local averages, that preserves the boundary condition in a
  relevant function space.

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