The Hagstrom–Warburton (HW) boundary operators play an important role in the development of high-order computational schemes for problems in unbounded domains. They have been used on truncating boundaries in the formulation of a sequence of high-order local Absorbing Boundary Conditions (ABCs) and in the Double Absorbing Boundary (DAB) method. These schemes proved to be very accurate, efficient, and generalizable for various wave equations and complex media. Yet, Finite Element (FE) formulations incorporating such high-order ABCs or DAB lack symmetry and positivity. As a result, they suffer from some deficiencies, namely (a) they do not allow the use of an explicit time-stepping scheme, since the global mass matrix and/or damping matrix are non-symmetric, and lumping is unsafe, (b) their stability is difficult to control under certain conditions, and (c) they render the fully-discrete problem non-symmetric even if the original problem in the unbounded domain is self adjoint, hence prevent the use of a symmetric algebraic solver. In this paper the HW-ABC for the scalar wave equation is applied to 2D waveguide configurations. It is manipulated in such a way that it leads to a symmetric FE-ABC formulation with positive definite matrices. The new symmetric formulation is achieved by applying a number of operations to the HW condition: first, combining each pair of recursive relations into one relation, then using the wave equation for each auxiliary function, and finally integrating the resulting ABC in time. The latter is the crucial step in the new method. The proposed method is free from the deficiencies (a)–(c) mentioned above. The DAB method undergoes a similar treatment. In this case, one of the matrices is slightly asymmetric, but deficiencies (a) and (b) are still prevented. The stability and accuracy of the new formulations are discussed, and their performance is demonstrated via numerical examples.

中文翻译：

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二维波导具有高阶吸收边界条件的对称有限元方案


Hagstrom-Warburton （HW） 边界算子在无界域中问题的高阶计算方案的开发中发挥着重要作用。它们已用于截断边界、制定一系列高阶局部吸收边界条件 （ABC） 和双吸收边界 （DAB） 方法。事实证明，这些方案非常准确、高效，并且适用于各种波动方程和复杂介质。然而，包含此类高阶 ABC 或 DAB 的有限元 （FE） 公式缺乏对称性和正性。因此，它们存在一些缺陷，即 （a） 它们不允许使用显式的时间步进方案，因为全局质量矩阵和/或阻尼矩阵是非对称的，并且集总是不安全的，（b） 它们在某些条件下难以控制它们的稳定性，以及 （c） 它们使完全离散问题成为非对称的，即使无界域中的原始问题是自伴随的， 因此，请避免使用对称代数求解器。在本文中，标量波动方程的 HW-ABC 应用于 2D 波导配置。它的处理方式是导致具有正定矩阵的对称 FE-ABC 公式。新的对称公式是通过对 HW 条件进行大量运算来实现的：首先，将每对递归关系组合成一个关系，然后对每个辅助函数使用波动方程，最后对得到的 ABC 进行时间积分。后者是新方法中的关键步骤。所提出的方法没有上述 （a）–（c） 的缺陷。DAB 方法也经历了类似的处理。 在这种情况下，其中一个矩阵略微不对称，但仍可以防止缺陷 （a） 和 （b）。讨论了新公式的稳定性和准确性，并通过数值示例证明了它们的性能。