An Efficient and Accurate Spectral Method for Acoustic Scattering in Elliptic Domains

An efficient and accurate method for solving the two-dimensional Helmholtz equation in domains exterior to elongated obstacles is developed in this paper.The method is based on the so called transformed field expansion(TFE) coupled with a spectral-Galerkin solver for elliptical domain using Mathieu functions.Numerical results are presented to show the accuracy and stability of the proposed method.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such,the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs,leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

Preface

This is a focused issue dedicated to the memory of the late Professor Ben-yu Guo(1942-2016),a prominent numerical analyst at Shanghai University and Shanghai Normal University,and a prolific researcher with more than 300 peer-reviewed publications,many of which are in prestigious journals.His work has been well recognized in the world and extensively cited.He received numerous prestigious awards,including a degree of Doctor of Science honoris causa from Sanford University in England.

Some Recent Advances on SpectralMethods for Unbounded Domains

We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi,Laguerre and Hermite functions.A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out,both theoretically and computationally.A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.

Efficient Dual Algorithms for Image Segmentation Using TV-Allen-Cahn Type Models

Variational image segmentation based on the Mumford and Shah model[31],together with implementation by the piecewise constant level-set method(PCLSM)[26],leads to fully nonlinear Total Variation(TV)-Allen-Cahn equations.The commonlyused numerical approaches usually suffer from the difficulties not only with the nondifferentiability of the TV-term,but also with directly evolving the discontinuous piecewise constant-structured solutions.In this paper,we propose efficient dual algorithms to overcome these drawbacks.The use of a splitting-penalty method results in TVAllen-Cahn type models associated with different"double-well"potentials,which allow for the implementation of the dual algorithm of Chambolle[8].Moreover,we present a new dual algorithm based on an edge-featured penalty of the dual variable,which only requires to solve a vectorial Allen-Cahn type equation with linear∇(div)-diffusion rather than fully nonlinear diffusion in the Chambolle’s approach.Consequently,more efficient numerical algorithms such as time-splitting method and Fast Fourier Transform(FFT)can be implemented.Various numerical tests show that two dual algorithms are much faster and more stable than the primal gradient descent approach,and the new dual algorithm is at least as efficient as the Chambolle’s algorithm but is more accurate.We demonstrate that the new method also provides a viable alternative for image restoration.

A MULTI-DOMAIN SPECTRAL IPDG METHOD FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER

This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann （DtN） boundary con- dition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treat- ment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on dif- ferent subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.

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.

A New Triangular Spectral Element Method II: Mixed Formulation and hp-Error Estimates

Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this paper.The method is based on equivalent first order system of the elliptic problem and rectangle-triangle transforms.It fully enjoys the ten-sorial structure and flexibility in handling complex domains by using nodal basis and unstructured triangular mesh.Different from the usual Galerkin formulation,the mixed form is particularly advantageous in this context,since it can avoid the singularity in-duced by the rectangle-triangle transform in the calculation of the matrices,and does not require the evaluation of the stiffness matrix.An hp a priori error estimate is pres-ented for the proposed method.The implementation details and some numerical exam-ples are provided to validate the accuracy and flexibility of the method.

On Spectral Approximations by Generalized Slepian Functions

We introduce a family of orthogonal functions,termed as generalized Slepian functions(GSFs),closely related to the time-frequency concentration problem on a unit disk in D.Slepian[19].These functions form a complete orthogonal system in L_(ωα)^(2)(−1,1)with̟ω_(α)(x)=(1−x)^(α),α>−1,and can be viewed as a generalization of the Jacobi polynomials with parameter(α,0).We present various analytic and asymptotic properties of GSFs,and study spectral approximations by such functions.