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 $\kappa$. 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 $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ and the spatial mesh width $h$. 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 $\kappa$-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 $L^2$- and $H^1$-error estimates are indeed optimal in $\kappa$ and $h$.
翻译:高κ情况下Ginzburg-Landau能量的离散最小化器误差界限
翻译后的摘要:
本研究针对有限元空间中Ginzburg-Landau能量的离散最小化器进行了研究。重点关注Ginzburg-Landau参数κ的影响。这个参数在物理学中非常重要,因为大的值会引发涡晶格的出现。由于涡旋必须在足够细的计算网格上解决,因此将κ的大小转化为网格分辨率条件非常重要,可以通过显式与κ和空间网格宽度h相关的误差估计来实现。为此,我们首先在一个离散空间的通用框架中进行了研究,其中我们以问题自适应的κ加权范数呈现收敛结果。然后,我们将结果应用于拉格朗日有限元和特定的广义有限元结构。在数值实验中,我们证实了我们推导出的L2和H1误差估计确实是在κ和h方面最优的。