Boundary Value Problems In Domains Research Articles

Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses

masses of diameter O(eδ) , are located near the boundary at distances of the order of O(δ) from each other, where δ = δ(e) → 0 . We pose the Dirichlet condition (respectively, the Neumann condition) on the parts of the boundary ∂Ω that are tangent (respectively, lying outside) the concentrated masses. We estimate the deviations of the solutions of the limit (averaged) problems from the solutions of the original problems in the norm of the Sobolev space W 1 2 for m< 2.

Inverse boundary value problems in domains with several obstacles

We consider the inverse boundary value problem for the Schrödinger equations with electromagnetic potentials in domains with several obstacles. We show that the boundary data allow one to recover integrals of potentials along broken rays. Then we prove the uniqueness modulo a gauge transform of the recovery of electromagnetic potentials from the integrals over broken rays under some geometrical conditions on the obstacles.

P‐version of the generalized FEM using mesh‐based handbooks with applications to multiscale problems

AbstractIn this paper, we analyse the p‐convergence of a new version of the generalized finite element method (generalized FEM or GFEM) which employs mesh‐based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the partition of unity method (PUM). We show that the p‐version of our GFEM is capable of achieving very high accuracy for multiscale problems which may be impossible to solve using the standard FEM. We analyse the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. We illustrate the robustness of the method by employing as model problem the Laplacian in a domain with a large number of closely spaced voids. Similar robustness can be expected for problems of heat‐conduction and elasticity set in domains with a large number of closely spaced voids, cracks, inclusions, etc. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Linear elliptic boundary value problems in varying domains

AbstractThe paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.

Natural Doundary Element Method for Elliptic Boundary Value Problems in Domains with Concave Angle

In this paper the natural boundary reduction for some elliptic boundary value problems with concave angle domains and its natural boundary element methods are investigated. Natural integral equations and Poisson integral formulae are given. A finite element methods of natural integral equations are discussed in details. The convergence of approximate solutions and their error estimates are obtained. Some numerical experiments are presented to demonstrate the performance of the method and our estimates. As an application, we present the coupling of FEM and natural boundary element.

Boundary value problems in domains with non Lipschitz boundary

Boundary value problems in domains with non Lipschitz boundary

SINGULAR INTEGRAL EQUATIONS IN TWO-DIMENSIONAL PROBLEMS OF THE THEORY OF CRACKS

We present a brief survey of the results of investigations devoted to the application of the method of singular integral equations to the solution of two-dimensional problems of fracture mechanics and carried out at the Karpenko Physicomechanical Institute of the Ukrainian National Academy of Sciences. Special attention is given to integral equations on piecewise smooth closed or nonclosed contours appearing in the boundary-value problems for domains with corners. We propose a new method for the solution of equations of this sort taking into account the singularities of stresses at angular points. The modified singular integral equations obtained as a result have continuous kernels and right-hand sides, i.e., belong to the same type of equations as in the case of smooth contours.

Completeness of Structural Solutions in Boundary-Value Problems for Domains of Special Form

General structures of solutions (GSS) of mathematical physics boundary-value problems in domains containing thin cuts, small cracks, hollows, etc. are considered. It is shown that in such cases, the classic L. V. Kantorovich GSS leads to incompleteness of structure formulas. New complete structure formulas are proposed for the domains.

Non-self-adjoint elliptic problems with a polynomial property in domains possessing cylindrical outlets to infinity

Solvability of mixed boundary-value problems in domains with cylindrical outlets to infinity is investigated for a certain class of non-self-adjoint differential operator-matrices., The structure of these matrices is such that the corresponding asymmetric quadratic forms possess a polynomial property, i.e., they degenerate only on finite-dimensional lineals of vector polynomials. With the help of elementary algebraic operations, this property enables one to indicate some attributes of boundary-value problems, namely, to calculate the operator index, to describe the kernel and co-kernel of an operator, to find an asymptotic behavior of solutions at infinity, etc. Bibliography: 16 titles.

The structure theorem for the Eulerian derivative of shape functionals defined in domains with cracks

Shape sensitivity analysis of boundary value problems in domains with cracks is important for applications; we refer the reader to the review [1] for some applications in fracture mechanics and references. In the simplest case such results are derived in [2]; we refer also to [3] and [4]. Since the structure theorem was not established, the direct approach is used in [2] which, in the case of the energy functional, r~luires existence of the shape derivative of solutions to elliptic equations in domains with cracks. The same approach is factually used in [5] in the case of unilateral conditions on the crack faces; however, in such a case the shape differentiability result does not seem to be known in the literature; we refer the reader to [6] for related results on the shape differentiability of solutions to variational inequalities in smooth domains. The problem with unilateral conditions on the crack faces is considered in [7] for a scalar equation and in [8] for an elasticity system, where the so-called Rice-Cherepanov formula is derived. It is shown in [7, 8] that the result on shape differentiability of solutions to variational inequalities is not required for the proof of differentiability of the energy functional with respect to the crack's length for problems with unilateral conditions prescribed on the crack faces.

A Calculus of Boundary Value Problems in Domains with Non–Lipschitz Singula Points

The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.

A Calculus of Boundary Value Problems in Domains with Non-Lipschitz Singula Points

The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.

Homogenization of boundary value problems in domains with rapidly oscillating nonperiodic boundary

Homogenization of boundary value problems in domains with rapidly oscillating nonperiodic boundary

The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary

We study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain Ωε⊂Rnwith a very rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter ε is posed on the oscillating part of the boundary and that the (n−1)-dimensional volume of this part goes to infinity as ε→0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem.

The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

The operator of a boundary value problem with Chaplygin-Zhukovskii-Kutta type conditions on an edge of the boundary has the fredholm property

UDC 517.9 w Introduction In the linear theory of thin wing (see [1-3] etc.), the Laplace equation for the velocity potential (I) is supplied with the usual boundary conditions as well as with the requirement of boundedness of the gradient grad (I) of the velocity potential on the trailing edge F of the wing. The possibility of satisfying this condition is associated with partial indefiniteness of the Dirichlet data on a part of the boundary, namely, they involve the unknown flow circulation whose values coincide with the trace of the potential on the line (edge) F. The requirement of boundedness of grad ~, which is attributed to Chaplygin, Zhukovskii, and Kutta, is mentioned in [4-6] and stated in [7]. In the two-dimensional situation, which has been studied comprehensively (see [1-9] etc.), F degenerates into a point and the operator of the problem with the Chaplygin-Zhukov skii-Kutta condition inherits the Fredholm property from the operator of the usual boundary value problem. As to the three-dimensional case, here the additional restrictions and arbitrariness do not fit in finite-dimensional spaces and thus can result in loss of the Fredholm property. So far no exhaustive mathematical investigation on the correct statement of the Chaplygin-Zhukov skii-Kutta conditions has been carried out (the author is aware of only one publication [10] where an attempt was made to use Sobolev spaces for this purpose, but the definitive answer was not given). In the present article we prove the Fredholm theorem for a problem that models the actual one. Specifically, we consider the problem in a bounded domain (rather than in R 3) and prescribe the Chal~lyginZhukovskii-Kutta conditions on a smooth closed curve F (the trailing edge of the wing has corner points and ends). The problem is described completely in w The function space in which the operator of the problem has the Fredholm property consists of functions with asymptotics of a certain form, and the Chaplygin-Zhukov skii-Kutta condition is restated as the requirement that the coefficient in the asymptotic term inducing the square root singularities vanish. All this, including the statement of the theorem on the Fredholm property, is contained in w167 3 and 4. The proof of the theorem is carried out as follows: first, a model problem in R~ is investigated in w and then the procedure of freezing the coefficients is applied in w All constructions are devised for concrete equations, and their generalization is not a purpose of this work (conditions on submanifolds of the boundary were studied for some situations in [11-13], etc.). In this article, we use the results of [14-21] in the theory of boundary value problems in domains with piecewise smooth boundary; they are collected in the book [21], to which we give exact references. I express my deep gratitude to Prof. Meister and Prof. Hinder for useful discussions of the subject of this paper. w Statement of the Problem

A Transmission Problem in the Scattering of Electromagnetic Waves by a Penetrable Object

Layer-potential techniques are used to study a transmission problem arising in the scattering of electromagnetic waves by a penetrable object. The method proposed does not involve the use of the calculus of pseudodifferential operators and hence it can be applied in domains with very little regularity. The solutions are represented as a combination of a curl and a double curl of a single layer-potential operator. The work relies on the important harmonic-analysis tools developed in recent years to study boundary-value problems in domains with minimal regularity assumptions.

Comparison of several mesh refinement strategies near edges

This paper is concerned with several refinement techniques of finite element meshes for treating elliptic boundary value problems in domains with re-entrant edges and corners. A priori mesh grading is explained, and it is combined with the well-known adaptive finite element method. For two representative examples the numerically determined error norms are recorded, and the different strategies are compared.

R-Functions in Boundary Value Problems in Mechanics

Described are the concepts and applications of the R-functions theory in continuum mechanics boundary value problems which model fields of different physical natures. With R-functions there appears the possibility of creating a constructive mathematical tool which incorporates the capabilities of classical continuous analysis and logic algebra. This allows one to overcome the main obstacle which hinders the use of variational methods when solving boundary value problems in domains of complex shape with complex boundary conditions, this obstacle being connected with the construction of so-called coordinate sequences. In contrast to widely used methods of the network type (finite difference, finite and boundary elements), in the R-functions method all the geometric information present in the boundary value problem statement is reduced to analytical form, which allows one to search for a solution in the form of formulae called solution structures containing some indefinite functional components. A method of constructing solution structures satisfying the required conditions of completeness has been developed. The structural formulae include the left-hand sides of the normalized equations of the boundaries of the domains or their regions being considered, thus allowing one to change the solution structure expeditiously when changing the geometric shape. Given in the work is a definition of the basic class of R-functions, solution with their help of the inverse problem of analytical geometry (construction of equations of specified configurations); generalization of the Taylor-Hermite formulae for functional spaces in which points are represented by lines and surfaces; and construction of solution structures of some types of boundary value problems. Shown are the solutions of a number of concrete problems in these application fields with the use of the RL language and POLYE system.

Initial value problems in elasticity

heat conduction phenomena like initial-boundary value problems in linear thermoelasticity are described. Except for the last example A is a selfadjoint operator. To solve such problems one has to give a solution concept first. Afterwards one tries to get more specific knowledge of the solutions obtained. For instance one inquires about their regularity, or about their asymptotic behaviour for large times and proves the existence of wave and scattering operators. Problems of inverse scattering theory are of great mathematical and practical interest (data like initial values, boundary or medium shall be recovered from the reflected signals). To treat such problems one has to know, for instance, how the scattering operator is determined by the boundary and to invert this mapping. In the high frequency limiting case one obtains relatively simple formulae. In the following, however, I shall not touch on such questions. Rather, in the first part of the lecture I shall only briefly treat linear boundary value problems for exterior domains, and also with the restriction that the underlying medium is homogeneous outside a sufficiently large ball. Of course boundary value problems for domains with unbounded boundaries are also of great importance.