Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

Monotone-iterative method for mixed boundary value problems for generalized difference equations with “maxima”

In this paper the authors investigate special type of difference equations which involve both delays and the maximum value of the unknown function over a past time interval. This type of equations is used to model a real process which present state depends significantly on its maximal value over a past time interval. An appropriate mixed boundary value problem for the given nonlinear difference equation is set up. An algorithm, namely, the monotone iterative technique is suggested to solve this problem approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima”, and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. Several numerical examples are considered to illustrate the practical application of the suggested algorithm.     

Discrete non-local boundary conditions for split-step Padé approximations of the one-way Helmholtz equation

This paper deals with the efficient numerical solution of the two-dimensional one-way Helmholtz equation posed on an unbounded domain. In this case, one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of the so-called discrete transparent boundary conditions for state-of-the-art parabolic equation methods, namely a split-step discretization of the high-order parabolic approximation and the split-step Padé algorithm of Collins. Finally, several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach.     

On the parallel version of the successive approximation method for quasilinear boundary value problem

The convergence properties of the successive approximation method to solve a quasilinear two points boundary value problem is studied. The successive approximation method is used to solve the parallel/multiple version of the problem. Conditions which assure the convergence of the method and error bound are given.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

A note on the determinant formulas computation of generalized inverse matrix Padé approximation

In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Padé approximation of type [n/2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.     

The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

l)ThisworkwassupportedbyNWOthroughgrantIBo7-3Go12.BOUNDAarv^LUEPRoBLEMFORELLIPTICEQUMIONwiTHMIXEDBOUNDAavCONDITION1.IntroductionInthispedwesketchavarietyofspecialmethodswhichareusedforconstructinge-unifornilyconvergelltschemes-WeshaJldemonstrateamethodwhichachieveshaprovedaccuracyforsolvingsingularlyperturbedb0undaryvalueproblemforeiliPicequatiouswithparabolicboundarylayers-InSecti0n4weshallintroduceanaturalclass,B,oftritefferenceschemes,inwhich(bytheabovementi0nedaP…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods

An approximate method based on piecewise linearization is developed for the determination of periodic orbits of nonlinear oscillators. The method is based on Taylor series expansions, provides piecewise analytical solutions in three-point intervals which are continuous everywhere and explicit three-point difference equations which are P-stable and have an infinite interval of periodicity. It is shown that the method presented here reduces to the well-known Störmer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padé approximation, another P-stable technique whenever the Jacobian is different from zero. The method is generalized for single degree-of-freedom problems that contain the velocity, and (approximate) analytical solutions are presented. Finally, by introducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Padé approximation, the method has been generalized to multiple degree-of-freedom problems and results in explicit three-point finite difference equations which only involve vector multiplications.     

非线性积分微分方程组奇摄动边值问题

本文讨论含积分算子的非线性微分方程组Ｒｏｂｉｎ边值问题的奇摄动,在适当假设条件下通过对角化技巧,利用逐步逼近法证明了解的存在,并得到直到Ｏ（ε＾Ｎ＋１）的按范数界限的一致有效估计。     

Smooth boundary integration method in a linear two-dimensional problem of incompressible fluid and solid

This paper presents a new application of a theoretical and computational method of smooth boundary integration which belongs to the methods of boundary integral equations. Smooth integration is not a method of approximation. In its final analytical form, a smooth-kernel integral equation is computerized easily and accurately.

Smooth integration is associated with a “pressure-vorticity” formulation which covers linear problems in elasticity and fluid mechanics. The solution presented herein is essentially the same as that reported in an earlier paper for regular elasticity. The constraint of incompressibility does not cause difficulties in the pressure-vorticity formulation.

The linear fluid mechanics problem formulated and solved in this paper covers Stokes' problem of a slow viscous flow, and has a wider interpretation. The translational inertia forces are incorporated in the linear problem, as in Euler's dynamic theory of inviscid flow. The centrifugal inertia forces are left for the non-linear problem. The linear problem is perceived as a step in solution of the non-linear problems.     

Method of successive approximations in the nonlinear theory of viscoelasticity

A method of successive approximations, a generalization of the Il'yushin method of elastic solutions, is proposed for solving problems of the nonlinear theory of elasticity in which the stress-strain relation is given in the form of a time operator Frechet-differentiable in a neighborhood of zero. The nonlinear relaxation kernels are found from the given nonlinear creep kernels for the principal quadratic theory of elasticity. These relations make it possible to formulate the boundary value problem for this theory. By way of illustration the problem of the pressure exerted on a space by a sphere is examined within the framework of the developed theory. The question of the convergence of the method is discussed in relation to the quadratic theory of visco-elasticity.Presented at the Third All-Union Conference on Theoretical and Applied Mechanics, Moscow (January, 1968).Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 236–242, March–April, 1969.     

Approximate calculation method for one-dimensional gas flows induced by nonmonotonic motion of a piston : PMM vol. 40, n 6, 1976, pp. 1058–1064

The problem of one-dimensional piston which at the beginning moves with increasing velocity into a gas at rest, then is decelerated, and finally stops, is solved by means of special series. The gas flow field is constructed by a successive joining of three characteristic Cauchy problems in terms of their characteristic solutions. Generalized solution of the problem of instantaneous arrest of the piston is derived. Obtained equations are used for the approximate calculation of the motion of generated shock waves.Representation of solutions of certain boundary value problems for nonlinear equations of the hyperbolic kind in the form of special series was proposed in [1, 2], The problem of the piston moving into a gas at rest is solved there, and the obtained solution was used for an approximate determination of the generated shock wave. The piston velocity was assumed to be monotonically increasing. That problem is solved here with the use of similar series in the case when the piston velocity is nonmonotonous,Numerical methods make it possible at present to determine one-dimensional flows similar to that considered below, and multidimensional problems can be solved by the method proposed in [1, 2]. The use of the proposed scheme for solving the problem of the multidimensional piston, whose velocity is nonmonotonous, does not present theoretical difficulties, but except that the formulas are more cumbersome.     

一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.