Exponential convergence of spectral approximations and time-domain computation of scattering problems.

This thesis consists of two parts. The first part is on rigorous error analysis of exponential convergence of orthogonal polynomial approximations under analytic assumption. The second part is on time-domain computation of scattering problems governed by wave equations and time-dependent Maxwell's equations. The spectral method employs global orthogonal polynomials or Fourier complex exponentials as basis functions, so it enjoys high-order accuracy (with only a few basis functions), if the underlying function is smooth (and periodic in the Fourier case). The typical algebraic order of convergence (i.e., $O(n^{-r})$, where $n$ is the number of polynomial basis functions, and $r$ is related to the Sobolev-regularity of the underlying function), is well documented in various monographs on spectral methods. It is widely accepted that if the function under consideration is sufficiently smooth or analytic, the convergence rate is of exponential order $O(q^n)$ (for constant $0<q<1$). However, there appear significantly less investigation and very limited discussions of such error bounds (mostly mentioned, but not proved). In the first part of the thesis, we provide error estimates for Jacobi polynomial expansions and Jacobi-Gauss-type interpolation of analytic functions (characterized by Bernstein ellipse $\mathcal E_\rho$ with $\rho>1$). More precisely, we derive error bounds of the type: $C(n)M \rho^{-n}$ with $\rho>1,$ where $M$ is the maximum value of the analytic function on the Bernstein ellipse, and where we specify the explicit dependence of $C(n)$ on $n$ and involved parameters. Moreover, the bounds are valid for small $n,$ and are significantly tighter than existing ones. We also discuss superconvergence of spectral interpolation. Indeed, the study of superconvergence phenomenon for $h$-version methods has had a great impact on scientific computing, especially on {\em a posteriori} error estimates and adaptive methods. With a belief that the scientific community would also benefit from the study of superconvergence phenomenon of spectral methods, some interesting results were obtained by Zhang. Here, we consider general Jacobi-Gauss-type interpolation and address the implication of superconvergence points identified from interpolating analytic functions, to functions with limited regularity, that is, the leading term of the interpolation error vanishes, but there is no gain in order of convergence, which is in distinctive contrast with analytic functions. The second part of this thesis is devoted to time-domain computation of wave scattering problems. It is known that the time-domain simulations are capable of capturing wide-band signals and modeling more general material inhomogeneities and nonlinearities, compared with the frequency-domain approaches e.g., the time-harmonic Helmholtz problems. We focus on fast and accurate computation of the exact circular/spherical nonreflecting boundary conditions (NRBCs) for acoustic wave equations and Maxwell's equations, which are used to truncate the scattering problems originally set in unbounded domains. However, a longstanding issue with the NRBC is its nonlocality in both time and space, due to the involvement of the temporal convolution and Fourier/spherical harmonic (or vector spherical harmonic in the Maxwell's case) series. We derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with $O(N_t)$ operations over $N_t$ successive time steps. In particular, for the Maxwell's equations, we derive a new formulation of the NRBC, which allows for the use of an analytic method for computing the involved inverse Laplace transform as in the acoustic case. We also propose efficient spectral solvers for the truncated problems with regular scatterers to show the accuracy and efficiency of this analytic-numerical approach.DOCTOR OF PHILOSOPHY (SPMS