Approximation of Exterior Problems. Optimal Conditions for the Laplacian.

Назаров, С. А.; Specovius–Neugebauer, Maria

doi:10.1524/anly.1996.16.4.305

Cited by 16 publications

(26 citation statements)

References 6 publications

(8 reference statements)

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“…For the most part, these methods are equivalent. In the required generality, the problem about a small cavity or inclusion was studied with the help of one or the other method in [37,38,9,39], and simple examples were considered in [37,40,8,41] and other publications. Therefore, the result of the asymptotic analysis of the energy functional (2.8) will be formulated below without accompanying calculations and the justification procedure, but with only brief explanations in Remarks 6.1 and 6.2.…”

Section: An (N × N )-Matrix Amentioning

confidence: 99%

The Eshelby theorem and patch optimization problem

Назаров

2010

St. Petersburg Math. J.

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Abstract. Let Ω 0 be an ellipsoidal inclusion in the Euclidean space R n . It is checked that if a solution of the homogeneous transmission problem for a formally selfadjoint elliptic system of second order differential equations with piecewise smooth coefficients grows linearly at infinity, then this solution is a linear vector-valued function in the interior of Ω 0 . This fact generalizes the classical Eshelby theorem in elasticity theory and makes it possible to indicate simple and explicit formulas for the polarization matrix of the inclusion in the composite space, as well as to solve a problem about optimal patching of an elliptical hole. §1. Setting of the transmission problem Let Ω 0 be a nonempty domain in the Euclidean space R n , n ≥ 2, with a smooth (of class C ∞ ) boundary, and let Ω 1 = R n \ Ω 0 be its exterior. Suppose it has compact closure:We place the origin of a Cartesian coordinate system x = (x 1 , . . . , x n ) in the interior of Ω 0 . Let D(∇ x ) be an (N ×k)-matrix of first order differential operators with constant and, in general, complex coefficients, and let A i , i = 0, 1, be numerical Hermitian positive definite matrices of size N × N . We introduce the formally selfadjoint (k × k)-matrices of second order differential operatorsand consider the following system of differential equations with transmission conditions on the interface Γ = ∂Ω 0 = ∂Ω 1 :

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Section: An (N × N )-Matrix Amentioning

confidence: 99%

The Eshelby theorem and patch optimization problem

Назаров

2010

St. Petersburg Math. J.

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“…Such approximations of unbounded problems by bounded ones are currently used (see e.g. [1,5,8] and references therein), and we must provide a condition on the boundary of the outer truncated domain.…”

Section: Practical Implementation -Truncation Of the Domainmentioning

confidence: 99%

“…These three boundary conditions are discussed for instance, in [8] (see also the discussion in [1] for the Helmholtz equation) where it is shown that for the Laplace equation with a right hand side which is non compactly supported, the Neumann boundary conditions might give a solution which does not converge as the outer boundary goes to infinity and that the mixed boundary condition is better than the Dirichlet.…”

Section: Practical Implementation -Truncation Of the Domainmentioning

confidence: 99%

“…We also remark that the mixed boundary condition adapted to our problem (which takes into account the decay of the solution in 1 R 2 instead of the classical 1 R decay for such problems) is not as natural to write as the one studied in [8] ∂φ ∂n 3 (see next section), there is no obvious manner to cancel this first term of the expansion with the normal derivative…”

Section: Practical Implementation -Truncation Of the Domainmentioning

confidence: 99%

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Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method

Alouges

2001

ESAIM: COCV

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Abstract. This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the decay of the solution at infinity and giving a rigidity matrix that can be very efficiently stored. All aspects are described here, from the practical construction of the mesh, the storage of the matrix, the error estimation of the method, the boundary conditions and a simple preconditionning technique. At the end of the paper, a typical computation of a uniformly magnetized ball is done and compared to the analytic solution. This method gives a natural alternatives to boundary elements methods for 3D computations.Mathematics Subject Classification. 65G99, 65Y20, 75Z05, 78A30, 78M10.

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“…Their choice is based on the asymptotic behavior of solutions at infinity. In particular, for elliptic boundary value problems in exterior domains and domains with cylindrical or conical outlets to infinity, ABCs in differential form were systematically developed during the last decades (see e.g., [1,2,4,5,7,9,10,14,23,24,32,34]) and the papers quoted there.) The common feature of local ABCs are estimates for the truncation error of the form u ∞ − u R = O(R −γ ) as R tends to infinity, with some γ > 0.…”

Section: Introductionmentioning

confidence: 99%

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Назаров¹,

Specovius–Neugebauer²

2008

Z. Anal. Anwend.

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Artificial boundary conditions are presented to approximate solutions to Stokes-and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v ∞ , p ∞ to the problems in the unbounded domainare the approximating solutions on the truncated domain Ω R , the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincaré operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R −N ), where N can be arbitrarily large.

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Approximation of Exterior Problems. Optimal Conditions for the Laplacian.

Cited by 16 publications

References 6 publications

The Eshelby theorem and patch optimization problem

The Eshelby theorem and patch optimization problem

Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

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