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Error bounds for discrete minimizers of the {G}inzburg--{L}andau energy in the high-κ\kappa regime

Abstract

In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter κ\kappa. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of κ\kappa into a mesh resolution condition, which can be done through error estimates that are explicit with respect to κ\kappa and the spatial mesh width hh. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted κ\kappa-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived L2L^2- and H1H^1-error estimates are indeed optimal in κ\kappa and hh

Similar works

This paper was published in KITopen.

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