一个扩散问题的自然边界元法与有限元法组合

本文讨论由Helmholtz方程描述的扩散问题的自然边界元法与有限元法的组合．取一个圆作为公共边界，用Fourier展开建立边界积分方程，将无界区域上的问题化为有界区域上的非局部边值问题．在变分方程中公共边界上的未知量只包含函数本身而不包含其法向导数，从而减少了未知数的数目，并且边界元剐度矩阵只有极少量不同的元素，有利于数值计算．这种组台方法优越于建立在直接边界元法基础上的组合方法．文中证明了变分解的唯一性，数值解的收敛性和误差估计．最后讨论了数值技术并给出一个算倒．     

On the coupling of BEM and FEM for exterior problems for the Helmholtz equation

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with FEM, and the normal derivative of the unknown function at the common boundary does not appear. The solvability of the variational equation and the error estimate are also discussed.

     

THE NUMERICAL SOLUTION OF FIRST KIND INTEGRAL EQUATION FOR THE HELMHOLTZ EQUATION ON SMOOTH OPEN ARCS

1. IntroductiouThe mathewtical tratod of the scattering Of theharmonic acoustic or electromagnoticwaves by an Mtely lOng sethecylindrical obstacle with a 8mooth opeu coDtour crewSeCtboF C Rs Ieads to unbounded boundare wtue problems for the Helmhltz equabo I3lwith wave nUmer h > 0.In the singtelayer Woach one Seeks the solutbo in the formwhere d8. is the element of arc length, and the fundamental solUbo to the Helmholtz equatfonis giveu byin terms Of the Hds fUnction H6') of order zero…     

A mixed boundary-value problem for the wave equation in a stratified medium for high-frequency oscillations

A boundary-value problem for the wave equation in a stratified medium with mixed boundary conditions on the boundary in the case of high oscillation frequencies is considered. The Helmholtz equation for a velocity function increasing monotonically with depth is investigated. The problem is reduced to an integral equation in the high-frequency approximation, and an explicitly smooth term of its asymptotic solution is constructed.     

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

A nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems

The article proposes a nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems. The free boundary value problem is broken up into two problems on nonoverlapping regions. In one region the problem is treated as a partial differential equation, while in the second region that contains the free boundary part, a variational inequality is considered. By solving these two related problems successively, we have shown that the successive solutions converge to the solution of the original problem. Application to a free surface seepage problem is given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The exterior Robin problem for the Helmholtz equation

A new method is given for solving the exterior Robin problem for the Helmholtz equation. The problem is reformulated as a new integral equation which is continuous as the field point approaches the boundary. It is shown that its solution can be represented as a convergent Neumann series for convex surfaces, for small values of the wave number. Examples are included which illustrate the method.     

Alternating iteration and elliptic variational inequalities

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.     

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

In this article we compute numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green’s function, and lead to a boundary element discretization. The Green’s function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green’s function and for a benchmark resonance problem are shown.     

Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.     

Fundamental Solution of Dirichlet Boundary Value Problem of Axisymmetric Helmholtz Equation

Fundamental solution of Dirichlet boundary value problem of axisymmetric Helmholtz equation is constructed via modi?ed Bessel function of the second kind, which uni?ed the formulas of fundamental solution of Helmholtz equation, elliptic type Euler-Poisson-Darboux equation and Laplace equation in any dimensional space.     

A class of random variational inequalities and simple random unilateral boundary value problems - existence,discretization, finite element approximation

The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation     

Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One‐step boundary knot method for discontinuous coefficient elliptic equations with interface jump conditions

This study makes the first attempt to apply the boundary knot method (BKM), a meshless collocation method, to the solution of linear elliptic problems with discontinuous coefficients, also known as the elliptic interface problems. The additional jump conditions are usually required to be prescribed at the interface which is used to maintain the well‐posedness of the considered problem. To solve the problem efficiently, the original governing equation is first transformed into an equivalent inhomogeneous modified Helmholtz equation in the present numerical formulation. Then the computational domain is divided into several subdomains, and the solution on each subdomain is approximated using the BKM approach. Unlike the conventional two‐step BKM, this study presents a one‐step BKM formulation which possesses merely one influence matrix for inhomogeneous problems. Several benchmark examples with various discontinuous coefficients have been tested to demonstrate the accuracy and efficiency of the present BKM scheme. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1509–1534, 2016     

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free‐space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null‐field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second‐part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

We consider a nonconforming hp -finite element approximation of a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H¹ as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved.     

The Helmholtz equation in a quadrant with Robin's conditions

The boundary value problem for the Helmholtz equation in a quadrant with different Robin's or impedance‐type conditions is considered by the use of variational methods and boundary pseudodifferential equations. The coerciveness of the underlying sesquilinear form in dependence of the impedance parameters is proved. The equivalence between the boundary value problem and a scalar uniquely solvable Wiener–Hopf–Hankel equation is obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solutions of some parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we introduce a variational form by defining a new function. Both continuous and discrete Galerkin procedures are illustrated in this paper. The error estimates are also derived.     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.