A Solve-Then-Discretize Paradigm for Spectral Methods

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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.