Advancements in the Ultraspherical Spectral Method

真实世界的流体流动和翼型问题的数值解需要一种精确、灵活和完全自适应的谱元方法。所谓的超球面谱方法，其稀疏性和正则性保持离散化，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谱方法的显着特性，目的是为偏微分方程生成高质量和工业强度的谱元素求解器。一个关键的特点将是它的鲁棒性捏边界功能，典型的翼型，这将减轻目前的巨大负担，网格生成算法。该项目将从根本上改变谱方法在计算数学和工程界的看法，广泛证明，如果仔细做，他们可以是一个灵活的，通用的，强大的数值工具。今天的伪谱方法提供方便和频谱精确的离散微分方程的解决方案。然而，他们导致密集的离散化，数值不稳定，并严重限制简单的几何形状。新颖的超球面谱方法是一种替代方法，它保留了相同的精度和方便性，但导致几乎带状的良好条件离散化，忠实地保留了基础微分算子的规律性，同时也适合于专门的快速线性代数例程。基于这种新的谱方法，PI将推导出一种新的完全自适应谱元方法。关键的新的计算特征将包括：（1）高精度的网格单元，是独立的纵横比;（2）真正的hp适应性，允许基本上任意大的单元度p和小的平均网格单元大小h（3）可灵活地求解各种具有一般边界约束的微分方程;（4）局部加密和网格粗化以解决角点奇异性问题。这种新的谱元方法将被应用到具有挑战性的偏微分方程的国家的最先进的数值模拟对流主导的流体流动问题。

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.