投影延拓法：一种复域的谱精确方法
The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains
摘要：
谱方法的一个基本组成部分是为测试空间和试验空间选择合适的基。在复杂的域上，这些基很难设计，需要使用域划分技术，如谱元法。在这项研究中，我们介绍了投影扩展（PE）方法，一种方法，产生光谱精确的解决方案，各种复杂的几何形状的问题，而不需要区域分解。这种技术建立在扩展方法所使用的见解，如浸入式边界平滑扩展和平滑力迫扩张（SFE）方法，旨在提高浸入式边界方法的精度。特别是，它耦合一个精确的扩展程序，功能上的任意域，无论连通性或规律性，与最小二乘最小化的边界条件。由此产生的程序是稳定的迭代应用和简单的推广到更高的维度。此外，它快速，稳健地产生指数收敛的解决方案，一些具有挑战性的测试问题，包括椭圆，抛物，牛顿流体流动，粘弹性问题。
英文摘要：
An essential ingredient of a spectral method is the choice of suitable bases for test and trial spaces. On complex domains, these bases are harder to devise, necessitating the use of domain partitioning techniques such as the spectral element method. In this study, we introduce the Projection Extension (PE) method, an approach that yields spectrally accurate solutions to various problems on complex geometries without requiring domain decomposition. This technique builds on the insights used by extension methodologies such as the immersed boundary smooth extension and Smooth Forcing Extension (SFE) methods that are designed to improve the order of accuracy of the immersed boundary method. In particular, it couples an accurate extension procedure, that functions on arbitrary domains regardless of connectedness or regularity, with a least squares minimization of the boundary conditions. The resulting procedure is stable under iterative application and straightforward to generalize to higher dimensions. Moreover, it rapidly and robustly yields exponentially convergent solutions to a number of challenging test problems, including elliptic, parabolic, Newtonian fluid flow, and viscoelastic problems.
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