EFFICIENT SPECTRAL METHODS FOR EIGENVALUE PROBLEMS OF THE INTEGRAL FRACTIONAL LAPLACIAN ONABALLOFANYDIMENSION

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis.And a sharp estimate on the algebraic system's condition number is established which behaves as N4s with respect to the polynomial degree N,where 2s is the fractional derivative order.The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces.Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived.Meanwhile,rigorous error estimates of the eigenvalues and eigenvectors are ob-tained.Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

Study on Parallel Computing

In this paper, we present a general survey on parallel computing. The main contents include parallel computer system which is the hardware platform of parallel computing, parallel algorithm which is the theoretical base of parallel computing, parallel programming which is the software support of parallel computing. After that, we also introduce some parallel applications and enabling technologies. We argue that parallel computing research should form an integrated methodology of ＂architecture algorithm programming application＂. Only in this way, parallel computing research becomes continuous development and more realistic.

Jacobi-Sobolev Orthogonal Polynomialsand Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains

In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains.We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian.As a result,the“stiffness”matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N^(2))arithmetic operations.Moreover,the singular factor in a typical kernel function can be fully absorbed by the basis.With the aid of Fourier analysis,we can prove the convergence of the scheme.We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions.We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.