Scholarly Works, Interdisciplinary Center for Applied Mathematics (ICAM)

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Browsing Scholarly Works, Interdisciplinary Center for Applied Mathematics (ICAM) by Author "Gunzburger, Max D."

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    Issues related to least-squares finite element methods for the stokes equations
    Deang, Jennifer M.; Gunzburger, Max D. (Siam Publications, 1998-10)
    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 velocity-vorticity-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.
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    On the Lawrence-Doniach and Anisotropic Ginzburg-Landau models for layered superconductors
    Chapman, S. Jonathan; Du, Qiang; Gunzburger, Max D. (Siam Publications, 1995-02)
    The authors consider two models, the Lawrence-Doniach and the anisotropic Ginzburg-Landau models for layered superconductors such as the recently discovered high-temperature superconductors. A mathematical description of both models is given and existence results for their solution are derived. The authors then relate the two models in the sense that they show that as the layer spacing tends to zero, the Lawrence-Doniach model reduces to the anisotropic Ginzburg-Landau model. Finally, simplified versions of the models are derived that can be used to accurately simulate high-temperature superconductors.