Li-Lian Wang

Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces

Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces are investigated. Some results on orthogonal projections and interpolations are established. Explicit expressions describing the dependence of approximation results on the parameters of Jacobi polynomials are given. These results serve as an important tool in the analysis of numerous quadratures and numerical methods for differential and integral equations.

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes α, β>−1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

Generalized Jacobi functions and their applications to fractional differential equations

In this paper, we consider spectral approximation of fractional dif- ferential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly de- signed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value prob- lems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well- conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.

Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2- and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented.

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Greens functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multi-dimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

A WELL-CONDITIONED COLLOCATION METHOD USING A PSEUDOSPECTRAL INTEGRATION MATRIX ∗

In this paper, a well-conditioned collocation method is constructed for solving general

Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross

p

A Triangular Spectral Element Method Using Fully Tensorial Rational Basis Functions

th order linear differential equations with various types of boundary conditions. Based on a suitable Birkhoff interpolation, we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features: the condition number of the linear system is independent of the number of collocation points, and the underlying boundary conditions are imposed exactly. Moreover, the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative (at interior collocation points), which is therefore called the pseudospectral integration matrix (PSIM). We show that PSIM produces the optimal integration preconditioner and stable collocation solutions with even thousands of points.

Analysis of spectral approximations using prolate spheroidal wave functions

Hyperbolic cross approximations by some classical orthogonal polynomials/functions in both bounded and unbounded domains are considered in this paper. Optimal error estimates in proper anisotropic weighted Korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established. These fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high-dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions.

Integration processes of ordinary differential equations based on Laguerre-Radau interpolations

A rational approximation in a triangle is proposed and analyzed in this paper. The rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform. Optimal error estimates for the