Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions

In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.

Composite Laguerre-Legendre Pseudospectral Method for Exterior Problems

In this paper,we propose a composite Laguerre-Legendre pseudospectral method for exterior problems with a square obstacle.Some results on the composite Laguerre-Legendre interpolation,which is a set of piecewise mixed interpolations coupled with domain decomposition,are established.As examples of applications,the composite pseudospectral schemes are provided for two model problems.The convergence of proposed schemes are proved.Efficient algorithms are implemented.Numerical results demonstrate the spectral accuracy in space of this new approach.

INFINITE ELEMENT METHOD FOR THE EXTERIOR PROBLEMS OF THE HELMHOLTZ EQUATIONS

There are two cases of the exterior problems of the Helmholtz equation. If λ ≥ 0 the bilinear form is coercive, and if λ < 0 it is the scattering problem. We give a new approach of the infinite element method, which enables us to solve these exterior problems as well as corner problems. A numerical example of the scattering problem is given. [ABSTRACT FROM AUTHOR]

ON THE UNIQUENESS OF BOUNDARY INTEGRAL EQUATION FOR THE EXTERIOR HELMHOLTZ PROBLEM

From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated in this paper. It is proved that the Sommerfeld's condition at the infinity is changed so that it is suitable not only for the radiative wave but also for the absorptive wave when we use the boundary integral equation to describe the exterior Helmholtz problem. There fore, the total energy of the system is conservative. The mathematical dealings to guarantee the uniqueness are discussed based upon this explanation.

EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT VARIABLES FOR PLANE ELASTICITY PROBLEMS

The exact form of the exterior problem for plane elasticity problems was produced and fully proved by the variational principle.Based on this,the equivalent boundary integral equations(EBIE) with direct variables,which are equivalent to the original boundary value problem,were deduced rigorously.The conventionally prevailing boundary integral equation with direct variables was discussed thoroughly by some examples and it is shown that the previous results are not EBIE.

BOUNDARY INTEGRAL FORMULAS FOR ELASTIC PLANE PROBLEM OF EXTERIOR CIRCULAR DOMAIN

After the stress function and the normal derivative on the boundary for the plane problem of exterior circular domain are expanded into Laurent series, comparing them with the Laurent series of the complex stress function and making use of some formulas in Fourier series and the convolutions, the boundary integral formula of the stress function is derived further. Then the stress function can be obtained directly by the integration of the stress function and its normal derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function is convenient to be used for solving the elastic plane problem of exterior circular domain.

LARGE TIME BEHAVIOR OF SOLUTIONS TO NEWTONIAN FILTRATION EQUATIONS WITH SOURCES

This article is concerned with large time behavior of solutions to the Neumann or Dirichlet problem for a class of Newtonian filtration equations ｜x｜λ＋Эt^-Эu=div（｜x｜^k∨u^m）＋｜x｜λ＋ku^p with 0 〈 m 〈 1,p 〉 1,λ≥0, k ∈ R. An interesting phenomenon is that there exist two thresholds k∞ and kl for the exponent k, such that the critical Fujita exponent pc for p exists and is finite if k E （k∞, k1）, otherwise, pc is infinite or does not exist.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fromtheMFS approximation has a block circulant structure,it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space.Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

The Solutions with Prescribed Asymptotic Behavior for the Exterior Dirichlet Problem of Hessian Equations

In this paper,we consider the exterior Dirichlet problem of Hessian equationsσk(λ(D^(2)u))=g(x)with g being a perturbation of a general positive function at infinity.The existence of the viscosity solutions with generalized asymptotic behavior at infinity is established by the Perron’s method which extends the previous results for Hessian equations.By the solutions of Bernoulli ordinary differential equations,the viscosity subsolutions and supersolutions are constructed.

Mixed Spectral and Pseudospectral Methods for a Nonlinear Strongly Damped Wave Equation in an Exterior Domain

The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc,using Fourier and generalized Laguerre functions.As an example,we consider a nonlinear strongly damped wave equation.The mixed spectral and pseudospectral schemes are proposed.The convergence is proved.Numerical results demonstrate the efficiency of this approach.

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichlet or Neumann boundary condition in a three-dimensional unbounded domain, which is the outside domain of a prolate spheroid. We express the Poisson integral formula and natural integral operator in a series form explicitly. Thus the original problem is reduced to a boundary integral equation on a prolate spheroid. The variational formula for the reduced problem and its well-posedness are discussed. Boundary element approximation for the variational problem and its error estimates, which have relation to the mesh size and the terms after the series is truncated, are also presented. Two numerical examples are presented to demonstrate the effectiveness and error estimates of this method.

THE OVERLAPPING DOMAIN DECOMPOSITION METHODFOR HARMONIC EQUATION OVER EXTERIORTHREE-DIMENSIONAL DOMAIN

Presents a study which applied the overlapping domain decomposition method based on the natural boundary reduction to solve the boundary value problem of harmonic equation over domain. Methods to solve boundary value problems; Contraction factor for the domain; Results.

The Exterior Tricomi Problem for Generalized Mixed Equations with Parabolic Degeneracy

This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness and existence of solutions are proved by a new method.

A NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR EXTERIOR 3-D PROBLEM

Examines a nonoverlapping domain decomposition method based on the natural boundary reduction. Development of the D-N alternating algorithm; Studies the convergence of the D-N method for exterior spherical domain; Discussion of the discrete form of the D-N alternating algorithm.

Axisymmetric Stokes Exterior Problem and Its Numerical Computation

We establish variational formulation and prove the existence and uniqueness of the three dimen- sional axisymmetric Stokes exterior problem in weighted spaces. Error estimates and convergence for P2 - P0 elements with infinite element methods are also obtained. Numerical experiments are presented to verify the theoretical analysis.

COMPOSITE LEGENDRE-LAGUERRE APPROXIMATION IN UNBOUNDED DOMAINS

Examines the development of the composite legendre approximation in unbounded domains. Proof of the stability and convergence of a proposed scheme; Discussion of two-dimensional exterior problems; Error estimations.

Natural Boundary Element Method for ParabolicEquations in an Unbounded Domain

In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using natural boundary reduction.The governing equation is first discretized in time, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier ex-pansion, we derive a natural integral equation of the elliptic problem related to timestep and Poisson integral integral formula over exterior circular domain. Finite elementdiscretization of the natural integral equation is employed to solve this problem. Thecomputational aspects of this method are discussed. Numerical results are presented toillustrate feasibility and efficiency of our method.

Some progress in spectral methods Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.

EXACT NONREFLECTING BOUNDARY CONDITIONS FOR AN ACOUSTIC PROBLEM IN THREE DIMENSIONS

In this paper, nonreflecting artificial boundary conditions are considered for an acoustic problem in three dimensions. With the technique of Fourier decomposition under the orthogonal basis of spherical harmonics, three kinds of equivalent exact artificial boundary conditions are obtained on a spherical artificial boundary. A numerical test is presented to show the performance of the method.

THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS

In this paper, we investigate the coupling of natural boundary element and finite element methods of exterior initial boundary value problems for hyperbolic equations. The governing equation is first discretized in time, leading to a time-step scheme, where an exterior elliptic problem has to be solved in each time step. Second, a circular artificial boundary TR consisting of a circle of radius R is introduced, the original problem in an unbounded domain is transformed into the nonlocal boundary value problem in a bounded subdomain. And the natural integral equation and the Poisson integral formula are obtained in the infinite domain Ω2 outside circle of radius R. The coupled variational formulation is given. Only the function itself, not its normal derivative at artificial boundary TR, appears in the variational equation, so that the unknown numbers are reduced and the boundary element stiffness matrix has a few different elements. Such a coupled method is superior to the one based on direct boundary element method. This paper discusses finite element discretization for variational problem and its corresponding numerical technique, and the convergence for the numerical solutions. Finally, the numerical example is presented to illustrate feasibility and efficiency of this method.