Dirichlet-Neumann Operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving-boundary problems, such as the classical water wave problem (free-surface ideal fluid flow under the influence of gravity and capillarity), to be restated in terms of interfacial quantities, which not only eliminates the boundary tracking problem, but also reduces the problem dimension. While these DNOs have been the object of much recent study regarding their numerical simulation and rigorous analysis, they have yet to be examined in the setting of laterally quasiperiodic boundary conditions. The purpose of this contribution is to begin this investigation with a particular eye towards the problem of more realistically simulating two and three dimensional surface water waves. Here we not only carefully define the DNO with respect to these boundary conditions for Laplace's equation, but we also show the rigorous analyticity of these operators with respect to sufficiently smooth boundary perturbations. These theoretical developments suggest a novel algorithm for the stable and high-order simulation of the DNO, which we implement and extensively test.

翻译：Dirichlet-Neumann算子（DNOs）在工程与科学领域中众多关键模型的建立、分析与模拟中具有重要作用。例如，这类算子使得移动边界问题——如经典的波浪问题（重力与表面张力作用下的自由表面理想流体流动）——能够用界面量重新表述，这不仅消除了边界追踪问题，还降低了问题的维度。尽管近年来针对这些DNO的数值模拟与严格分析已有大量研究，但在横向准周期边界条件下的情形尚未得到充分探讨。本文旨在启动这一方向的研究，并特别关注更真实地模拟二维与三维表面水波的问题。我们不仅针对拉普拉斯方程在此类边界条件下精确定义了DNO，还严格证明了这些算子关于充分光滑边界摄动的解析性。这些理论进展启发了一种稳定且高阶模拟DNO的新算法，我们实现了该算法并进行了广泛测试。