Integral equations in the wave scattering problem at an irregular interface of domains

Modeling the scattering of electromagnetic waves at an interface of media with different characteristics, one encounters the conjugation problem. Using the method of boundary integral equations and the theory of generalized potentials, we prove the classical resolvability of this problem. The boundary is assumed to be irregular. This means that the plane is divided into two domains by a curve which coincides with a straight line, except for a finite part, producing the irregularity. We propose algorithms for the approximate solution of the conjugation problem based on the spline methods for the solution of integral equations. We theoretically substantiate the computational scheme, namely, we prove the convergence and estimate the convergence rate.     

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

Numerical conformal mapping methods based on Faber series

Methods are presented for approximating the conformal map from the interior of various regions to the interior of simply-connected target regions with a smooth boundary. The methods for the disk due to Fornberg (1980) and the ellipse due to DeLillo and Elcrat (1993) are reformulated so that they may be extended to other new computational regions. The case of a cross-shaped region is introduced and developed. These methods are used to circumvent the severe ill-conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast Fourier methods available on the disk. The methods are based on expanding the mapping function in the Faber series for the regions. All of these methods proceed by approximating the boundary correspondence of the map with a Newton-like iteration. At each Newton step, a system of linear equations is solved using the conjugate gradient method. The matrix-vector multiplication in this inner iteration can be implemented with fast Fourier transforms at a cost of O(N log N). It is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly. Several computational examples are given along with a discussion of the accuracy of the methods.     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

A smoothing homotopy method for variational inequality problems on polyhedral convex sets

In this paper, based on the Robinson’s normal equation and the smoothing projection operator, a smoothing homotopy method is presented for solving variational inequality problems on polyhedral convex sets. We construct a new smoothing projection operator onto the polyhedral convex set, which is feasible, twice continuously differentiable, uniformly approximate to the projection operator, and satisfies a special approximation property. It is computed by solving nonlinear equations in a neighborhood of the nonsmooth points of the projection operator, and solving linear equations with only finite coefficient matrices for other points, which makes it very efficient. Under the assumption that the variational inequality problem has no solution at infinity, which is a weaker condition than several well-known ones, the existence and global convergence of a smooth homotopy path from almost any starting point in $R^n$ are proven. The global convergence condition of the proposed homotopy method is same with that of the homotopy method based on the equivalent KKT system, but the starting point of the proposed homotopy method is not necessarily an interior point, and the efficiency is more higher. Preliminary test results show that the proposed method is practicable, effective and robust.     

A non-overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two-dimension

We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two-dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non-overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.     

类Hartree-Fock方程的数值方法

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系（含几十到几百个电子）的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法（ACE）,投影的C-DⅡS方法（PC-DⅡS）方法,以及插值可分密度近似方法（ISDF）,我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.     

Existence results for operator equations in abstract spaces and an application

In this paper we apply the recursion method presented in (Heikkilä and Lakshmikantham, 1994) to derive fixed point theorems and existence results for operator equations in partially ordered sets. The obtained results, combined with related results of (Heikkilä, 2002) are then applied to operator equations in ordered Hilbert spaces and to a functional elliptic boundary value problem.     

An artificial boundary method for the Hull-White model of American interest rate derivatives

The Hull-White (HW) model is a widely used one-factor interest rate model because of its analytical tractability on liquidly traded derivatives, super-calibration ability to the initial term structure and elegant tree-building procedure. As an explicit finite difference scheme, lattice method is subject to some stability criteria, which may deteriorate the computational efficiency for early exercisable derivatives. This paper proposes an artificial boundary method based on the partial differential equations (PDEs) to price interest rate derivatives with early exercise (American) feature under the HW model. We construct conversion factors to extract the market information from the zero-coupon curve and then reduce the infinite computational domain into a finite one by using an artificial boundary on which an exact boundary condition is derived. We then develop an implicit θ-scheme with unconditional stability to solve the PDE in the reduced bounded domain. With a finite computational domain, the optimal exercise strategy can be determined efficiently. Our numerical examples show that the proposed scheme is accurate, robust to the truncation size, and more efficient than the popular lattice method for accurate derivative prices. In addition, the singularity-separating technique is incorporated into the artificial boundary method to enhance accuracy and flexibility of the numerical scheme.     

Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems

In this paper we present a new method for the solution of optimization problems with PDE constraints. It is based on simultaneous pseudo-time stepping for evolution equations. The new method can be viewed as a continuous reduced SQP method in the sense that it uses a preconditioner derived from that method. The reduced Hessian in the preconditioner is approximated by a pseudo-differential operator, whose symbol can be investigated analytically. We apply our method to a boundary control model problem. The new optimization method needs 3.2-times the overall computational effort of the solution of simulation problem alone.     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.     

On the existence of more positive solutions of periodic bvps with singularity

We consider Periodic boundary value problems for ordinary second order differential equations of the form u′′=f(t,u,u′), Where f satisfies the (local) Carathéodory conditions and can have a singularity in the second variable.Writing our problem in an operator can be computed on. These sets are not convex, in general. Using the degree theory we get at least one fixed point of the operator at each such set which leads to the existence and localization of more solutions of the related Periodic boundary value problem. Our results are based on the generalized lower and upper functions method from Rach?nková and Tvrdý[15].     

Global strong solutions of Navier-Stokes equations with interface boundary in three-dimensional thin domains

In this article, we study the spectrum of the Stokes operator in a 3D two layer domain with interface, obtain the asymptotic estimates on the spectrum of the Stokes operator as thickness ε goes to zero. Based on the spectral decomposition of the Stokes operator, a new average-like operator is introduced and applied to the study of Navier-Stokes equation in the two layer thin domains under interface boundary condition. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. This article is a continuation of our study on the Stokes operator under Navier friction boundary condition. Due to the viscosity distinction between the two layers, the Stokes operator displays radically different spectral structure from that under Navier friction boundary condition, then causes great difficulty to the analysis.     

Fast conformal mapping of an ellipse to a simply connected region

The iterative method of Wegmann (1978) for conformal mapping of a region E to a region G is reformulated in terms of a conjugation operator. If the boundary of G is sufficiently smooth, the method converges quadratically in a Sobolev space W. To overcome the crowding an elongated canonical region E should be chosen if the image region G is elongated. For ellipses E the operator of conjugation can be expressed in a very simple way in terms of Fourier series. Numerically, this can be performed very efficiently by fast Fourier transform (FFT).     

A parallel implementation of overlapping Schwarz method for the dual reciprocity method

We present a parallel algorithm for the overlapping domain decomposition boundary integral equation method for two dimensional partial differential equations. In addition to the improvement of the ill-conditioning and the computational efficiency achieved by domain partitioning, using a parallel computer with p processors can offer up to p times efficiency. Assuming direct solution is used throughout, partitioning the domain into p subregions and employing a processor for each subproblem, overall, result in p² times efficiency over using a single domain and a single processor, taking into account that a sequential algorithm of the underlying method can improve the computational efficiency at least p times over using a single domain. Some numerical results showing the efficiency of the parallel technique will be presented.     

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.