Abstract. Least-squares finite element methods have become increasingly popular for the approximate solution of first-order systems of partial differential equations. Here, after a brief review of some existing theories, a number of issues connected with the use of such methods for the velocityvorticity-pressure formulation of the Stokes equations in two dimensions in realistic settings are studied through a series of computational experiments. Finite element spaces that are not covered by existing theories are considered; included in these are piecewise linear approximations for the velocity. Mixed boundary conditions, which are also not covered by existing theories, are also considered, as is enhancing mass conservation. Next, problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved. A conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh-dependent weights in the least-squares functional almost always improves the accuracy of the approximations. Concluding remarks concerning three-dimensional problems, the nonlinear Navier-Stokes equations, and the conditioning of the discrete systems are provided. Key words. least squares, finite element methods, Stokes equations AMS subject classification. 65N30PII. S10648275952945261. Introduction. Least-squares finite element methods have always held out the attraction of yielding discrete linear systems that are symmetric and positive definite even for problems for which other methods, e.g., mixed finite element methods, fail to do so; see, e.g. , [2]-[48], [50]-[56], [58], and [60]-[84]. In many settings such as the primitive variable formulation of the Stokes equations, these methods suffer from two serious problems. The first is that conforming discretizations require the use of continuously differentiable finite element functions and the second is that the condition number of the discrete equations is often proportional to h −4 , where h denotes some measure of the grid size. However, least-squares finite element methods have recently been receiving increasing attention in both the engineering and mathematics communities; see, e.g., [3] [84]. The focus of this attention has been on the application of least-squares finite element methodologies to first-order systems of partial differential equations for which one can, in principle, use merely continuous finite element functions and for which one may often prove that the condition numbers of the discrete systems are proportional to h −2 . The mathematical references cited above consider least-squares finite element methods in idealized situations, i.e., for problems having simple boundary conditions and smooth solutions and in the asymptotic limit of the grid size measure h → 0. Unfortunately, these are usually far from true in most applications of the methods to practical problems; indeed, there are many such settings for which mathematical

Vortices in superconductors are tubes of magnetic flux, or equivalently, cylindrical current loops, that penetrate into a material sample. Knowledge about the structure and dynamics of collections of vortices is of importance both to the understanding of the basic physics of superconductors and to the design of devices. We first discuss homogeneous isotropic superconductors that can be modelled by the Ginzburg-Landau theory. We then discuss variants of this model that can account for inhomogenieties and anisotropies due to impurities, thickness variations, and thermal fluctuations. These all effect changes in the vortex state, as do changes in the applied magnetic field and current strengths and directions. Through computational simulations, we use the various models to illustrate these changes. In particular, we examine the pinning of vortices by thickness variations in thin-films, by impurities and by grain boundaries, the effects that changes in the thickness of a simple Josephson junction have on the structure of the vortex state, transitions that occur in the vortex state as the applied magnetic field is increased, and distortions of that state due to thermal fluctuations.

In anisotropic superconductors having an arbitrary orientation of the sample surface relative to the crystal principal axes, the surface critical field H c3 is less than 1.695H c2 unless the field is situated along one of the principal crystal planes. Below H c3 in the vicinity of nucleation, the order parameter scales as ͱH c3 ϪH.Computational studies for infinite cylinders having rectangular cross sections are presented which show that, due to corners and a finite cross section, the surface superconductivity state persists for fields above the theoretically predicted value for semi-infinite samples. They also show that vortices exist within the surface superconductivity sheath above the bulk critical field.

Thermal fluctuations and randomly distributed defects in superconductors are modeled by stochastic variants of the time-dependent Ginzburg-Landau equations. Numerical simulations are used to compare the effects of additive and multiplicative noise models.

It is well known that thermal fluctuations and material impurities affect the motion of vortices in superconductors. These effects are modeled by variants of a time-dependent Ginzburg-Landau model containing either additive or multiplicative noise. Numerical computations are presented that illustrate the effects that noise has on the dynamics of vortex nucleation and vortex motion. For an additive noise model with relatively low variances, it is found that the vortices form a quasi-steadystate lattice in which the vortex core sizes remain roughly fixed but their positions vibrate. Two multiplicative noise models are considered. For one model having relatively long-range order, the sizes of the vortex cores vary in time and from one vortex to another. Finally, for the additive noise case, we show that as the variance of the noise tends to zero, solutions of the stochastic time-dependent GinzburgLandau equations converge to solutions of the corresponding equations with no noise. c 2002 Elsevier Science (USA)