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Partial Differential Equations (PDEs)

The focus of the research is on designing accurate and efficient numerical methods for linear and nonlinear partial differential equations (PDEs). In particular, we focus on the numerical solution of PDEs by spectral methods. When the solution of an elliptic PDE is analytic, the error decays exponentially quickly. For time-dependent PDEs, most of the emphasis has been on low-order finite difference schemes for the time derivative and spectral schemes for the spatial derivatives. Our research examines space-time spectral methods which converge exponentially in both space and time. Spectral convergence and condition number estimates for canonical linear PDEs (heat, wave and Stokes) have been shown. Space-time spectral convergence has been demonstrated numerically for many nonlinear PDEs including the viscous Burgers, Cahn-Hilliard, KdV, nonlinear reaction-diffusion, Sine-Gordon, Navier-Stokes and MHD equations at high Reynolds numbers. Current work includes space-time spectral methods for PDEs on irregular domains, delay differential equations, and solution of resultant equations using machine learning.

Singular integral equations

Singular integral equations arise in the reformulation of elliptic partial differential equations where data is defined on prescribed boundaries of reduced dimensionality. The reduction of dimensionality and transformation from an elliptic partial differential equation into a singular integral equation arises naturally from Green’s representation theorem.

Singular integral equations have a rich history in scattering problems for electromagnetics and seismic imaging, fracture mechanics, fluid dynamics, and beam physics. For applications including random matrix theory, asymptotics of orthogonal polynomials, and integrable systems, singular integral equations arise via reformulation as Riemann–Hilbert problems.

In this group, we develop a new class of fast, stable, and well-conditioned spectral methods for singular integral equations. The combination of direct solvers with hierarchical solvers allows numerical simulations with domains consisting of thousands of disjoint boundaries with millions of degrees of freedom. The use of a hierarchical solver as a pre-conditioner in a parallel iterative solver will extend this to new problems involving millions of disjoint boundaries. To extend the spectral method to more complicated geometries, we develop fast and stable algorithms for transforming expansion coefficients in Chebyshev bases to more exotic polynomial bases. As the spectral method is extended to three-dimensional elliptic partial differential equations, new fast and stable algorithms for transforming expansion coefficients will be required and new applications will also be explored including cloaking, scattering from fractal antennae, and scattering in parabolically stratified media such as optical fibers.


Fast orthogonal polynomial transforms

Orthogonal polynomials enable the best numerical approximation and computation with functions in weighted L^2 spaces. As such, they are a fundamental tool in the design and analysis of spectral methods for the solution of ordinary and partial differential and integral equations. In this group, we investigate novel methods to accelerate the operations of synthesis and analysis (evaluating a truncated orthogonal polynomial series on a particular grid and approximating the coefficients in such an expansion given function values on the grid) as well as the connection problem to perhaps more familiar polynomials such as those of Jacobi, Laguerre, and Hermite. The connection problem allows for flexible algorithms for associated, gyroscopic, rationally modified, semi-classical, and Sobolev orthogonal polynomials, among many others.