An efficient, accurate and robust solution technique for variable coefficient elliptic partial differential equations in complex geometries

数值模拟在科学发现和设备开发中起着关键作用，因为它们能够降低测试理论和想法的成本。这些模拟通常涉及物理模型规定的问题的解决。在许多情况下，提出问题的几何是复杂的，和/或建模物理现象的方程导致几何复杂性。涉及复杂几何形状的应用包括材料设计、逆散射和流体模拟。由于解决方案的复杂性，我们希望尽可能降低计算成本。做到这一点的关键是使用尽可能少的自由度来捕捉物理，并将其与有效的解决技术相结合。最近的一种称为层次庞加莱-斯特克洛夫（HPS）方法的数值技术能够解决许多问题。该方法已被证明是有效的高频散射问题，并已集成到反散射模拟。然而，HPS方法的当前版本无法处理大多数应用程序中出现的复杂几何形状。这项研究将解决这一不足，使该方法能够应用于复杂的几何形状。这项工作还将把新版本的HPS方法与现有的复杂几何形状软件连接起来，使其能够集成到模拟软件包中。本课题所考虑的数值模拟是含高频亥姆霍兹问题的变系数线性椭圆偏微分方程。这些方程出现在散射、静电等物理现象的建模中，以及使用许多时间步进技术解决时间相关问题时（即在流体模拟中）。当前版本的HPS方法只能处理易于从正方形或立方体映射的几何图形。这是有问题的，因为在大多数应用程序中，存在不属于这些类别的几何特征。该项目将HPS方法扩展到一般的几何形状，并将与现有的网格生成软件无缝集成。此外，该项目将使HPS方法有效地解决三维问题（目前还不是），并发展必要的分析来支持在实践中观察到的数值结果。由于HPS方法的鲁棒性，它可以在已知先验计算成本的情况下用于材料设计和逆散射。这意味着从业者可以自信地将这种技术用于散射应用，因为他们知道该方法达到了期望的准确性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Numerical simulations play a key role in scientific discovery and device development because they have the ability to reduce the cost of testing theories and ideas. These simulations often involve the solution of problems that are prescribed by physics models. In many cases, the geometry where the problem is posed is complex and/or the equation that is modeling the physical phenomena results in geometry complexities. Applications involving complex geometries include materials design, inverse scattering and fluid simulations. Due to the sophisticated nature of solutions, it is desirable to keep the computational cost as low a possible. The key to doing this is to use as few degrees of freedom as possible to capture the physics and to couple this with efficient solution techniques. A recent numerical technique called the hierarchical Poincare-Steklov (HPS) method is able to do this for many problems. This method has been demonstrated to be effective for high frequency scattering problems and has been integrated into inverse scattering simulations. However, the current version of the HPS method is not able to handle the complex geometries that arise in most applications. This research will address this shortfall allowing the method to be applied to complex geometries. This work will also connect the new version of the HPS method to existing software for complex geometries allowing it to be integrated into simulation packages.The numerical simulations under consideration in this project are linear elliptic partial differential equations with variable coefficients including high frequency Helmholtz problems. These equations arise in the modeling of physical phenomena such as scattering, electrostatics, and when using many time-stepping techniques for solving time dependent problems (i.e. in fluid simulations). The current version of the HPS method can only handle geometries that can be easily mapped from a square or cube. This is problematic as in most applications, there are geometric features that do not fall into either of these categories. This project will extend the HPS method to a general range of geometries and will integrate seamlessly with existing mesh generation software. Additionally, this project will make the HPS method efficient for three dimensional problems (which it currently is not) and develop the necessary analysis to support the numerical results that are observed in practice. Thanks to the robustness of the HPS method, it can be used in material design and inverse scattering with a known a priori computational cost. This means that practitioners can confidently use this technique for scattering applications knowing that the method is achieving the desired accuracy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.