论文标题
平滑强迫扩展方法：一种用于求解复杂域上椭圆方程的高级技术
The Smooth Forcing Extension Method: A High-Order Technique for Solving Elliptic Equations on Complex Domains
论文作者
论文摘要
用于在任意域上求解椭圆方程的高阶数值方法通常需要专门的机械，例如有限元素方法的高质量符合网格，以及边界积分方法的正交规则。这些工具使得将这些技术应用于更高的维度变得困难。相比之下，固定的笛卡尔网格方法（例如浸没边界（IB）方法）易于应用和推广，但通常是低阶精度。在这项研究中，我们介绍了平滑的强迫扩展（SFE）方法，这是一种固定的笛卡尔网格技术，它基于IB方法的见解，并允许人们获得任意的准确性顺序。我们的方法依靠一种新型的傅立叶延续方法来计算不均匀术语的扩展到任何所需的规律性。这与高度准确的非均匀快速傅立叶变换相结合，用于插值操作，以产生快速，健壮的方法。数值测试证实该技术在一维测试问题上的预期性能准确地执行。在较高的维度中，性能甚至更好，在某些情况下会产生子几何收敛。我们还展示了该技术如何应用于解决抛物线问题，并在一般域上计算椭圆算子的特征值，以说明其稳定性和对概括性的稳定性。
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.
