Fast spectral methods and their applications

谱方法是应用数学和科学计算中用于数值求解某些微分方程的一类技术。许多问题的解表现出局部奇异性，这影响了通常的光谱/谱元方法的准确性。许多表现出异常扩散的复杂系统可以用分数阶偏微分方程更好地建模，由于它们的非局部性质，这在数值上具有挑战性。光谱/谱元方法通常导致密集或块密集和病态矩阵，求解困难且昂贵。这个项目的重点是设计、分析和实现快速和鲁棒的谱方法来解决一类具有数值挑战性的问题。数值模拟将使我们能够以合理的CPU成本和周转时间处理具有严格精度和/或内存要求的挑战性问题，并将有助于通过快速准确的数值模拟更好地理解材料科学和流体动力学中的一些基本问题。该项目的另一个重要目标是让研究生学习计算和应用数学的必要技能，以便他们能够在科学和工程领域取得成功。PI将通过以下任务来解决这些问题：(i)开发有效的Muntz Galerkin方法来处理奇异解问题；（ii）通过构造具有广义Jacobi和Laguerre函数的特殊基函数，开发出求解一类分数阶微分方程的高效、准确的谱方法，并推导出相应的误差估计；（iii）开发具有最佳计算复杂度的直接结构化求解器，用于由谱/谱元离散化产生的密集或块密集线性系统；（iv）开发有效的光谱算法来求解铁电和多铁纳米结构中电磁耦合的相场模型。该研究将产生快速、稳定的一类偏微分方程的直接谱/谱元求解器，以及快速的雅可比/球谐变换。本项目还将为铁电和多铁纳米结构中电磁耦合相场模型的耦合非线性系统提供一套高效、精确求解的计算模块。

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. Solutions for many problems of interest exhibit local singular behaviors which contaminate the accuracy of usual spectral/spectral-element methods. Many complex systems exhibiting anomalous diffusion can be better modeled with fractional partial differential equations, which are numerically challenging due to their nonlocal nature. Spectral/spectral-element methods usually lead to dense or block dense and ill-conditioned matrices that are difficult and expensive to solve. The focus of this project is to design, analyze, and implement fast and robust spectral methods for a class of numerically challenging problems. The numerical simulations will enable us to handle challenging problems having stringent accuracy and/or memory requirements with a reasonable cost in CPU and turn-around time, and will contribute to a better understanding of some fundamental issues in materials science and fluid dynamics through fast and accurate numerical simulations. Another important goal of this project is to engage graduate students in learning necessary skills of computational and applied mathematics so that they can pursue a successful career in sciences and engineering.The PI will address these issues with the following tasks: (i) develop effective Muntz Galerkin method to deal with problems with singular solutions; (ii) develop efficient and accurate spectral methods for solving a class of fractional differential equations by constructing special basis functions with generalized Jacobi and Laguerre functions, and derive corresponding error estimates; (iii) develop direct structured solvers with optimal computational complexity for dense or block dense linear systems arising from spectral/spectral-element discretization; and (iv) develop efficient spectral algorithms for solving the phase-field model of electro-magnetic couplings in ferroelectric and multiferroic nanostructures. The research will result in fast and stable direct spectral/spectral-element solvers for a class of partial differential equations as well as fast Jacobi/spherical harmonic transforms. This project will also result in a set of computational modules to efficiently and accurately solve the coupled nonlinear system for the phase-field model of electro-magnetic couplings in ferroelectric and multiferroic nanostructures.