In this work, we present and analyze the numerical stability of two coupled finite element formulations. The first one is the h-a-formulation and is well suited for modeling systems with superconductors and ferromagnetic materials. The second one, the so-called t-a-formulation with thin-shell approximation, applies for systems with thin superconducting domains. Both formulations involve two coupled unknown fields and are mixed on the coupling interfaces. Function spaces in mixed formulations must satisfy compatibility conditions to ensure stability of the problem and reliability of the numerical solution. We propose stable choices of function spaces using hierarchical basis functions and demonstrate the effectiveness of the approach on simple 2D examples.

中文翻译：

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高温超导体混合有限元配方的稳定性

在这项工作中，我们提出并分析了两个耦合有限元公式的数值稳定性。第一个是 ha 公式，非常适合用于对具有超导体和铁磁材料的系统进行建模。第二个，即所谓的薄壳近似 ta 公式，适用于具有薄超导域的系统。两种公式都涉及两个耦合的未知场，并在耦合界面上混合。混合公式中的函数空间必须满足兼容性条件，以确保问题的稳定性和数值解的可靠性。我们提出了使用分层基函数的函数空间的稳定选择，并在简单的 2D 示例上证明了该方法的有效性。