Advancements in the Ultraspherical Spectral Method

实际流体流动和翼型问题的数值求解需要一种精确、灵活和完全自适应的谱元方法。所谓的超球面光谱方法，由于其稀疏性和保持规律性的离散化，有望克服许多传统的计算障碍。本研究项目将开发和研究超球面光谱方法的显著特性，旨在生产高质量和工业强度的偏微分方程光谱元求解器。一个关键的特征将是其对压缩边界特征的鲁棒性，典型的翼型，这将减轻目前对网格生成算法的巨大负担。该项目将从根本上改变计算数学和工程界对谱方法的看法，通过广泛地证明，如果做得仔细，谱方法可以是一种灵活、通用和强大的数值工具。今天的伪谱方法为微分方程的解提供了方便性和光谱精确的离散化。然而，它们导致密集离散化、数值不稳定和对简单几何的严重限制。新的超球面光谱方法是一种替代方法，它保留了相同的准确性和便利性，但导致几乎带状的良好条件离散化，忠实地保留了底层微分算子的规律性，同时也适用于专门的快速线性代数例程。基于这种新的谱元方法，PI将推导出一种新的基于数学的网格几何全自适应谱元方法。关键的新计算特征将包括：(1)独立于纵横比的网格元素高精度；(2)真正的hp自适应，允许基本上任意大的单元度p和小的平均网格单元尺寸h（不考虑病态）；(3)具有求解具有一般边界约束的各种微分方程的灵活性；(4)角点奇异点的局部细化和网格粗化。这种新的谱元方法将应用于具有挑战性的偏微分方程，用于最先进的对流主导流体流动问题的数值模拟。

The numerical solution of real-world fluid flow and airfoil problems needs an accurate, flexible, and fully-adaptive spectral element method. The so-called ultraspherical spectral method, with its sparsity and regularity preserving discretizations, is promising to overcome many of the traditional computational barriers. This research project will exploit and investigate the remarkable properties of the ultraspherical spectral method with the aim of producing a high quality and industrial-strength spectral element solver for partial differential equations. One key feature will be its robustness to pinching boundary features, typical with airfoils, that will alleviate the current tremendous burden on mesh generation algorithms. The project will radically alter the perception of spectral methods in the computational mathematics and engineering communities by extensively demonstrating that, when done carefully, they can be a flexible, general, and powerful numerical tool.Today's pseudospectral methods deliver both convenience and spectrally accurate discretizations for the solution of differential equations. However, they lead to dense discretizations, numerical instability, and a severe limitation to simple geometries. The novel ultraspherical spectral method is an alternative that retains the same accuracy and convenience, but leads to almost banded well-conditioned discretizations that faithfully preserves the regularity of the underlying differential operator while also being amenable to specialized fast linear algebra routines. Based on this new spectral method, the PI will derive a new mathematically-grounded fully-adaptive spectral element method for meshed geometries. Key novel computational features will include: (1) A high accuracy on mesh elements that is independent of the aspect ratio; (2) True hp-adaptivity that allows for essentially arbitrarily large element degree p and small average mesh element size h (without concern of ill-conditioning); and (3) The flexibility to solve a wide range of differential equations with general boundary constraints; and (4) Local refinement and mesh coarsening for the resolution of corner singularities. This new spectral element method will be applied to challenging partial differential equations for the state-of-the-art numerical simulation of advection-dominated fluid flow problems.