M. Ganesh

M. Ganesh, Q. T. Le Gia, I. H. Sloan

In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error analysis is based on the Gevrey regularity of the solutions of the Navier-Stokes equations on the sphere. The scheme is designed for convenient application of fast evaluation techniques such as the fast Fourier transform (FFT), and the implementation is based on a stable adaptive time discretization.

V. Dominguez, M. Ganesh

We propose, analyze, and implement interpolatory approximations and Filon-type cubature for efficient and accurate evaluation of a class of wideband generalized Fourier integrals on the sphere. The analysis includes derivation of (i) optimal order Sobolev norm error estimates for an explicit discrete Fourier transform type interpolatory approximation of spherical functions; and (ii) a wavenumber explicit error estimate of the order $\mathcal{O}(κ^{-\ell} N^{-r_\ell})$, for $\ell = 0, 1, 2$, where $κ$ is the wavenumber, $N$ is the number of interpolation/cubature points on the sphere and $r_\ell$ depends on the smoothness of the integrand. Consequently, the cubature is robust for wideband (from very low to very high) frequencies and very efficient for highly-oscillatory integrals because the quality of the high-order approximation (with respect to quadrature points) is further improved as the wavenumber increases. This property is a marked advantage compared to standard cubature that require at least ten points per wavelength per dimension and methods for which asymptotic convergence is known only with respect to the wavenumber subject to stable of computation of quadrature weights. Numerical results in this article demonstrate the optimal order accuracy of the interpolatory approximations and the wideband cubature.

V. Dominguez, M. Ganesh

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications.In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than $N$. The spherical interpolatory approximation is efficiently constructed by applying the FFT techniques with only ${\mathcal{O}}(N^2 \log N)$ complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.

M. Ganesh, S. C. Hawkins, C. Jeznach et al.

The free space Maxwell dielectric problem can be reduced to a system of surface integral equations (SIE). A numerical formulation for the Maxwell dielectric problem using an SIE system presents two key advantages: first, the radiation condition at infinity is exactly satisfied, and second, there is no need to artificially define a truncated domain. Consequently, these SIE systems have generated much interest in physics, electrical engineering, and mathematics, and many SIE formulations have been proposed over time. In this article we introduce a new SIE formulation which is in the desirable operator form identity plus compact, is well-posed, and remains well-conditioned as the frequency tends to zero. The unknowns in the formulation are three dimensional vector fields on the boundary of the dielectric body. The SIE studied in this paper is derived from a formulation developed in earlier work by some of the authors~\cite{ganesh2014all}. Our initial formulation utilized linear constraints to obtain a uniquely solvable system for all frequencies. The new SIE introduced and analyzed in this article combines the integral equations from \cite{ganesh2014all} with new constraints. We show that the new system is in the operator form identity plus compact in a particular functional space, and we prove well-posedness at all frequencies and low-frequency stability of the new SIE.