沙堆与超导体：非符合线性有限元近似用于混合形式的拟变分不等式
Sandpiles and Superconductors: Nonconforming Linear Finite Element Approximations for Mixed Formulations of Quasi-Variational Inequalities
AI总结：
本文提出基于非符合线性有限元的简化近似方法，用于解决沙堆和II型超导体中带梯度约束的混合形式拟变分不等式，并通过数值实验验证其有效性。
摘要：
相似的进化变分和拟变分不等式，带有梯度约束，在沙堆和II型超导体的建模中出现。最近，这些不等式的混合形式被用于建立拟变分不等式情况下的存在性结果。此类形式，这也是一个额外的优势，使得能够数值确定不仅原始变量，例如沙堆和超导体的演变表面和磁场，还能够确定对偶变量，即沙流量和电场。之前的工作中，这些混合形式的数值近似采用了最低阶的Raviart--Thomas元。这里，我们引入基于非符合线性有限元的这些混合形式的更简单的数值近似。我们证明了这些近似的（子序列）收敛性，并通过数值实验说明其有效性。
英文摘要：
Similar evolutionary variational and quasi-variational inequalities with gradient constraints arise in the modeling of growing sandpiles and type-II superconductors. Recently, mixed formulations of these inequalities were used for establishing existence results in the quasi-variational inequality case. Such formulations, and this is an additional advantage, made it possible to determine numerically not only the primal variables, e.g. the evolving sand surface and the magnetic field for sandpiles and superconductors, respectively, but also the dual variables, the sand flux and the electric field. Numerical approximations of these mixed formulations in previous works employed the Raviart--Thomas element of the lowest order. Here we introduce simpler numerical approximations of these mixed formulations based on the nonconforming linear finite element. We prove (subsequence) convergence of these approximations, and illustrate their effectiveness by numerical experiments.
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