A Solve-Then-Discretize Paradigm for Spectral Methods

谱方法是偏微分方程（PDE）数值求解的三大技术之一（沿着与有限差分和有限元方法），在流体流动和翼型模拟方面特别强大。 该研究项目旨在开发一个新的无限维框架来求解偏微分方程，以获得具有竞争力的计算算法，该算法保留了微分算子的连续结构，有望克服许多具有谱离散化的硬快速计算障碍。我们的目标是产生一个自适应的，强大的，工业强度的迭代求解器的光谱方法，以允许流体流动的准确分辨率的集合。我们还将开发工具，用于计算微分算子的伪谱和连续谱，促进非弹性散射的理解。这些结果将有助于证明，谱精确方法，如果仔细做，是灵活的，通用的，强大的数值工具，在计算数学和engineering.The标准的范式求解偏微分方程是首先离散方程，然后解决由此产生的线性系统。这种方法有一些缺点的谱方法有关的预处理器的设计，引入非正规性，和扰动的频谱。该项目正在开发的无限维框架通过避免微分算子的离散化来保持PDE的连续结构，而是仅离散光滑函数，例如PDE的解和源项。不与有限部分的微分算子承诺，使我们能够开发强大的Krylov为基础的迭代求解器，直接从微分算子激励预条件，计算连续部分的运营商的频谱，并制定了一个理论基础的解决方案和特征函数的自适应分辨率的基础上误差分析。我们将把这些新工具应用于对流主导的流体流动以及非弹性散射的数值模拟。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Spectral methods are one of the big three technologies (along with finite differences and finite element methods) for the numerical solution of partial differential equations (PDEs) and are particularly powerful for fluid flow and airfoil simulations. This research project aims to develop a new infinite-dimensional framework for solving PDEs to derive competitive computational algorithms that preserve the continuum structure of differential operators, promising to overcome many of the hard-and-fast computational barriers with spectral discretizations. We aim to produce a collection of adaptive, robust, and industrial-strength iterative solvers for spectral methods to allow for the accurate resolution of fluid flows. We will also develop tools for computing the pseudospectra and continuous spectra of differential operators, facilitating improved understanding of inelastic scattering. The results will help to demonstrate that spectrally-accurate methods, when done carefully, are flexible, general, and powerful numerical tools in computational mathematics and engineering.The standard paradigm for solving a PDE is to first discretize the equation and then solve the resulting linear system. This approach has a number of drawbacks for spectral methods related to the design of preconditioners, the introduction of non-normality, and the perturbation of spectra. The infinite-dimensional framework under development in this project preserves the continuum structure of PDEs by avoiding the discretization of differential operators, and instead only discretizes smooth functions, such as the solution and the source terms of the PDE. Not working with finite sections of differential operators promises to enable us to develop robust Krylov-based iterative solvers, motivate preconditioners directly from the differential operator, compute the continuous part of the spectrum of operators, and develop a theoretical foundation for the adaptive resolution of solutions and eigenfunctions based on error analysis. We will apply these new tools to the numerical simulation of advection-dominated fluid flow as well as inelastic scattering.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.