Fast spectral methods and their applications

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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.