Boundary-value problem in the correlative approximation of the method of quasi-periodic components for a unidirectional fiber composite

The boundary-value problem in the correlative approximation of the method of quasi-periodic components and a numerical algorithm based on the boundary element method for determining the nonuniform stress fields in the matrix of a unidirectional fiber composite with a disordered structure are considered. The numerical results and analysis of the probability density function, for example, for normal stresses at some points of the interface of absolutely rigid fibers of the composite are presented. Perm State Technical University, Russia. Translated from Mekhanika Kompozytnykh Materialov, Vol. 35, No. 5, pp. 629–642, September–October, 1999.     

Singularly perturbed Dirichlet boundary-value problem for a stationary system in the linear elasticity theory

We consider a singularly perturbed Dirichlet boundary-value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary-value problem to eigenelements of the corresponding limiting boundary-value problem, when the parameter ? which defines the diameter of the small cavity tends to zero.     

Boundary element-Landweber method for the Cauchy problem in linear elasticity

In this paper, an iterative algorithm based on the Landwebermethod in combination with the boundary element method is developedfor solving the Cauchy problem in isotropic linear elasticity.An efficient regularizing stopping criterion is also employed.The numerical results obtained confirm that the iterative methodproduces a convergent and stable numerical solution with respectto increasing the number of boundary elements and decreasingthe amount of noise added into the input data, respectively.     

Refined mixed finite element method for the elasticity problem in a polygonal domain

The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient λ are obtained for λ large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323–339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009     

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 posteriori error estimation for the dual mixed finite element method of the elasticity problem in a polygonal domain

In this article, we propose a residual based reliable and efficient error estimator for the new dual mixed finite element method of the elasticity problem in a polygonal domain, introduced by M. Farhloul and M. Fortin. With the help of a specific generalized Helmholtz decomposition of the error on the strain tensor and the classical decomposition of the error on the gradient of the displacements, we show that our global error estimator is reliable. Efficiency of our estimator follows by using classical inverse estimates. The lower and upper error bounds obtained are uniform with respect to the Lamé coefficient λ, in particular avoiding locking phenomena. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

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.     

The structure of the boundary element matrix for the three-dimensional Dirichlet problem in elasticity

The algebraic properties of the matrix arising for the three-dimensional Dirichlet problem for Lamé equations in a rotational domain by the boundary element method are considered. The use of the special basis leads to a matrix having a block structure with sparse blocks. The possible strategies for the efficient solution of the above problem are discussed. © 1998 John Wiley & Sons, Ltd.     

On the structure of the spectrum for the elasticity problem in a body with a supersharp spike

We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity O(r ^1+γ), where r is the distance to the vertex of the spike and γ > 1 is the sharpness exponent.     

Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two‐dimensional bounded domain with thin shoots, depending on a small parameter ε. Under the assumption that the width of the shoots goes to zero, as ε tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order ε, we construct the two‐term asymptotics of the eigenvalues of the problem, as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.     

A mathematical analysis of the Gumbel test for jumps in stochastic volatility models

This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Hölder continuous with an arbitrary random Hölder exponent and we have no restrictions concerning an additional drift term. Therefore, for example, we are allowing for long and short-range dependence. In the case of existing additive jumps, we give divergence results in a general semimartingale setting and investigate the speed of divergence depending on the jump activity. As a by-product of our analysis we also deduce an optimal pathwise estimator for the spot volatility process. Moreover, we provide a detailed simulation study that compares the power of the Gumbel test with the power of the jump test proposed by Barndorff–Nielsen and Shephard in 2006 for Hölder exponents close to zero. Finally, both tests are applied to a real dataset.     

On the solvability of a nonlinear boundary value problem arising in the theory of growing cell populations

In this paper we establish some results regarding the existence of solution on L₁ spaces to a nonlinear boundary value problem originally proposed by Lebowitz and Rubinow (J. Math. Biol. 1974; 1 :17–36) to model an age‐structured proliferating cell population. Our approach, based on topological methods, uses essentially the specific properties of weakly compact sets on L₁ spaces. Our results provide positive answers to the questions posed in Jeribi (Nonlinear Anal. Real World Appl. 2002; 3 :85–105). Copyright © 2005 John Wiley & Sons, Ltd.     

A Riemannian variant of the Fletcher–Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata

In this paper, we focus on the stochastic inverse eigenvalue problem with partial eigendata of constructing a stochastic matrix from the prescribed partial eigendata. A Riemannian variant of the Fletcher–Reeves conjugate gradient method is proposed for solving a general unconstrained minimization problem on a Riemannian manifold, and the corresponding global convergence is established under some assumptions. Then, we reformulate the inverse problem as a nonlinear least squares problem over a matrix oblique manifold, and the application of the proposed geometric method to the nonlinear least squares problem is investigated. The proposed geometric method is also applied to the case of prescribed entries and the case of column stochastic matrix. Finally, some numerical tests are reported to illustrate that the proposed geometric method is effective for solving the inverse problem.     

Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. It turns out that the existence or not of such solutions depends on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter, we prove the existence of solutions to this boundary value problem. For large values of the same parameter, we prove the nonexistence of solutions. We also provide rigorous bounds for the values of this parameter, which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions applied to an associated two‐point boundary value problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order

This article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. The sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method, meantime, the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given. Some results of previous literature are extended and improved. A numerical example is also included to illustrate the effectiveness of the proposed results.     

在局部区域上的奇摄动非线性方程Robin边值问题解的渐近性态

本文是讨论了一类在局部区域上的奇摄动非线性方程Ｒobin边值问题,利用泛函数分析及算子理论,得到了相应问题解的渐近性态。     

A short note on the paper “Convergence of the TAGE iterative method for the system arisen from the cubic spline approximation for the solution of two-point BVPs with forcing function in integral form”, by Mohanty, Jain and Dhall

In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036-3047] and then correct it.