The smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains
The smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains
复制标题
平滑力扩展法：求解复杂域上椭圆方程的高阶技术
DOI：
10.1016/j.jcp.2021.110390
复制
发表时间：
2021
影响因子：
4.1
通讯作者：
Griffith, Boyce E.
中科院分区：
文献类型：
--
作者：
Qadeer, Saad;Griffith, Boyce E.
求解任意区域上椭圆方程的高阶数值方法通常需要专门的机器，例如有限元方法的高质量协调网格和边界积分方法的求积规则。这些工具使得将这些技术应用于更高维度变得困难。相比之下，固定笛卡尔网格方法，如浸没边界（IB）方法，易于应用和推广，但通常是低阶精度。在这项研究中，我们介绍了光滑强迫扩展（SFE）方法，一个固定的笛卡尔网格技术，建立在IB方法的见解，并允许一个获得任意阶的精度。我们的方法依赖于一种新的傅立叶延拓方法来计算扩展的非齐次项的任何所需的规律性。这与用于插值操作的高精度非均匀快速傅立叶变换相结合，以产生快速且鲁棒的方法。数值试验证实，该技术精确地执行一维测试问题的预期。在更高的维度，性能甚至更好，在某些情况下产生子几何收敛。我们还演示了如何将这种技术可以应用于解决抛物问题和计算椭圆算子的特征值一般域，在说明其稳定性和顺应性的推广过程。
High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.