Over the past decade, partial differential equation models in elliptical geometries have become a focus of interest in several scientific and engineering applications: the classical studies of flow past a cylinder, the spherical particles in nano-fluids and spherical water filled domains are replaced by elliptical geometries which more accurately describe a wider class of physical problems of interest. Optimized Schwarz methods (OSMs) are among the best parallel methods for such models. We study here for the first time OSMs with elliptical domain decompositions, i.e. decompositions into an ellipse and elliptical rings. Using the technique of separation of variables, we decouple the spatial variables and reduce the subdomain problems to radial Mathieu like equations defined on finite intervals, which allows us to derive and study a new family of OSMs. Our analysis reveals that the optimized transmission parameters are not constants any more along the elliptical interfaces. We can prove however also that using the constant optimized parameters from the straight interface analysis in the literature scaled locally by the interface curvature is still efficient in an asymptotic sense, which leads to the important discovery of a unique factor in the optimized parameters and asymptotic performance determined by the geometry of the decomposition. We use numerical examples to illustrate our analysis and findings.

在过去的十年中，椭圆几何形状的偏微分方程模型已成为一些科学和工程应用的关注点：经典的圆柱流研究，纳米流体和球形充水域中的球形颗粒被椭圆替换可以更准确地描述感兴趣的物理问题的几何形状。优化的Schwarz方法（OSM）是此类模型的最佳并行方法。我们在这里首次研究具有椭圆域分解的OSM，即分解为椭圆和椭圆环。使用变量分离技术，我们将空间变量解耦，并将子域问题简化为径向Mathieu，例如在有限区间上定义的方程，这使我们能够派生和研究OSM的新家族。我们的分析表明，优化的传输参数不再是沿椭圆界面的常数。然而，我们也可以证明，在文献中通过界面曲率局部缩放的直界面分析中使用恒定的优化参数在渐近意义上仍然有效，这导致对优化参数和渐近性能中唯一因素的重要发现。由分解的几何形状确定。我们使用数字示例来说明我们的分析和发现。然而，我们还可以证明，在渐近意义上，使用文献中通过界面曲率局部缩放的直接界面分析中的恒定优化参数仍然有效，这导致在优化参数和渐近性能中发现独特因素的重要发现由分解的几何形状确定。我们使用数字示例来说明我们的分析和发现。然而，我们也可以证明，在文献中通过界面曲率局部缩放的直界面分析中使用恒定的优化参数在渐近意义上仍然有效，这导致对优化参数和渐近性能中唯一因素的重要发现。由分解的几何形状确定。我们使用数字示例来说明我们的分析和发现。