We propose four approaches to solve time‐harmonic Maxwell's equations in ℝ³ through the finite element method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the multiple multipole program (MMP) in the unbounded complement. MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that spawn exact solutions of the homogeneous equations. Each of these sources is anchored at a point that, if singular, is placed outside the respective domain of approximation. In the MMP domain, we assume that material parameters are piecewise constant, which induces a partition into one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own MMP trial space. Hence, in addition to the transmission conditions between the FEM and MMP domains, one also has to impose transmission conditions connecting the MMP subdomains. Coupling approaches arise from seeking stationary points of Lagrangian functionals that both enforce the variational form of the equations in the FEM domain and match the different trial functions across subdomain interfaces. We discuss the following approaches:

1. Least‐squares‐based coupling using techniques from PDE‐constrained optimization.
2. Discontinuous Galerkin coupling between the meshed FEM domain and the single‐entity MMP subdomains.
3. Multi‐field variational formulation in the spirit of mortar FEMs.
4. Coupling through the Dirichlet‐to‐Neumann operator.

中文翻译：

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耦合有限元和电磁波传播的辅助源

我们建议解决时谐麦克斯韦方程组在四种方法ℝ ³通过有限元方法（FEM）在包含参数不均匀性的有界区域中结合无界补集中的多极子程序（MMP）。MMP属于辅助源方法和Trefftz方法类，因为它使用点源来生成齐次方程的精确解。这些源中的每一个都锚定在一个点（如果是奇异点），位于相应的近似域之外。在MMP域中，我们假设材料参数是分段常数，这会导致划分为一个无界子域和其他有界但可能非常大的子域，每个子域都需要自己的MMP试用空间。因此，除了在FEM和MMP域之间的传输条件外，还必须强加连接MMP子域的传输条件。耦合方法是通过寻找拉格朗日函数的平稳点而产生的，该平稳点既可以在FEM域中强制执行方程的变分形式，又可以跨子域接口匹配不同的试验函数。我们讨论以下方法：

1. 使用PDE约束优化中的技术进行基于最小二乘的耦合。
2. 网格化有限元域和单实体MMP子域之间的不连续Galerkin耦合。
3. 遵循砂浆有限元分析法的多场变分公式。
4. 通过Dirichlet-to-Neumann算子进行耦合。