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。许多感兴趣的问题的解决方案表现出当地的奇异行为，污染通常的谱/谱元方法的准确性。分数阶偏微分方程可以更好地模拟许多表现出反常扩散的复杂系统，由于其非局部性质，这在数值上具有挑战性。谱/谱元方法通常导致稠密或块稠密和病态矩阵，这是困难和昂贵的解决。该项目的重点是设计，分析和实现快速和强大的谱方法的一类数值挑战性的问题。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。本计划的另一个重要目标是让研究生学习计算及应用数学的必要技能，以便他们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函数和Laguerre函数的特殊基函数，发展求解一类分数阶微分方程的高效精确的谱方法，并得到相应的误差估计;（iii）开发具有最佳计算复杂性的直接结构化求解器，用于由光谱/光谱元素离散化产生的密集或块密集线性系统;以及（iv）开发高效的光谱算法，用于求解铁电和多铁纳米结构中电磁耦合的相场模型。该研究将导致快速和稳定的直接谱/谱元求解一类偏微分方程以及快速雅可比/球谐变换。该项目还将产生一组计算模块，以有效和准确地解决铁电和多铁性纳米结构中电磁耦合的相场模型的耦合非线性系统。

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.