Results 1  10
of
32
Singularities of Electromagnetic Fields in Polyhedral Domains
 ARCH. RATIONAL MECH. ANAL
, 1997
"... In this paper, we investigate the singular solutions of time harmonic Maxwell equations in a domain which has edges and polyhedral corners. It is now well known that in the presence of nonconvex edges, the solution fields have no square integrable gradients in general and that the main singulariti ..."
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Cited by 31 (4 self)
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In this paper, we investigate the singular solutions of time harmonic Maxwell equations in a domain which has edges and polyhedral corners. It is now well known that in the presence of nonconvex edges, the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the Laplace operator. We show how this type of result can be derived from the classical Mellin analysis, and how this analysis leads to sharper results concerning the singular parts which belong to H 1 : For the generating singular functions, we exhibit simple and explicit formulas based on (generalized) Dirichlet and Neumann singularities for the Laplace operator. These formulas are more explicit than the results announced in our note [9].
Singularities of Maxwell interface problems
 M2AN Math. Model. Numer. Anal
, 1998
"... . We investigate time harmonic Maxwell equations in heterogeneous media, where the permeability ¯ and the permittivity " are piecewise constant. The associated boundary value problem can be interpreted as a transmission problem. In a very natural way the interfaces can have edges and corners. We giv ..."
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Cited by 30 (7 self)
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. We investigate time harmonic Maxwell equations in heterogeneous media, where the permeability ¯ and the permittivity " are piecewise constant. The associated boundary value problem can be interpreted as a transmission problem. In a very natural way the interfaces can have edges and corners. We give a detailed description of the edge and corner singularities of the electromagnetic fields. Introduction Physical objects interacting with electromagnetic waves not only tend to have corners and edges, but are frequently composed of several materials with different electric and magnetic properties. The electromagnetic fields then have singularities not only at the exterior corners and edges, but also at the singular points of the interfaces between the different materials. We show how these singularities can be analyzed using the classical Kondrat 'ev method [13]. In the paper [8], we studied the singularities at corners and edges of a homogeneous material. Here we continue this investigat...
Adjoints Of Elliptic Cone Operators
, 2001
"... We study the adjointness problem for the closed extensions of a general belliptic operator A 2 x Di m b (M ; E), > 0, initially dened as an unbounded operator A : C 1 c (M ; E) x L 2 b (M ; E) ! x L 2 b (M ; E), 2 R. The case where A is a symmetric semibounded operator is of par ..."
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Cited by 25 (8 self)
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We study the adjointness problem for the closed extensions of a general belliptic operator A 2 x Di m b (M ; E), > 0, initially dened as an unbounded operator A : C 1 c (M ; E) x L 2 b (M ; E) ! x L 2 b (M ; E), 2 R. The case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator. 1.
Weighted Regularization of Maxwell Equations in Polyhedral Domains
 Numer. Math
, 2001
"... We present a new method of regularizing time harmonic Maxwell equations by a divergence part adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usua ..."
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Cited by 19 (3 self)
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We present a new method of regularizing time harmonic Maxwell equations by a divergence part adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the nondensity of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient.
Adaptive Wavelet Schemes for Elliptic Problems  Implementation and Numerical Experiments
 SIAM J. Scient. Comput
, 1999
"... Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination o ..."
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Cited by 19 (11 self)
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Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briey review rst the main theoretical facts, give a detailed description of the algorithm, highlight the essential data structures and illustrate the results by one and two dimensional numerical examples.
Asymptotics of Arbitrary Order for a Thin Elastic Clamped Plate, II. Analysis of the Boundary Layer Terms
 Anal
, 1995
"... . This paper is the last of a series of two, where we study the asymptotics of the displacement in a thin clamped plate as its thickness tends to 0 . In Part I, relying on the structure at infinity of the solutions of certain model problems posed on unbounded domains, we proved that the combination ..."
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Cited by 16 (6 self)
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. This paper is the last of a series of two, where we study the asymptotics of the displacement in a thin clamped plate as its thickness tends to 0 . In Part I, relying on the structure at infinity of the solutions of certain model problems posed on unbounded domains, we proved that the combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multiscale asymptotics of the displacement and optimal estimates in energy norm. The "profiles" for the boundary layer terms are solutions of such model problems. In this paper, adapting SaintVenant's principle to our framework, we prove the results which we used in Part I. Investigating more precisely the structure of the boundary layer terms, we go further in the analysis performed in Part I: the introduction of edge layer terms along the intersections of the clamped face with the top and the bottom of the plate respectively, allows estimates in higher order norms. These edge layer ...
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
RESOLVENTS OF ELLIPTIC BOUNDARY PROBLEMS ON CONIC MANIFOLDS
, 2006
"... Abstract. We prove the existence of sectors of minimal growth for realizations of boundary value problems on conic manifolds under natural ellipticity conditions. Special attention is devoted to the clarification of the analytic structure of the resolvent. 1. ..."
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Cited by 7 (2 self)
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Abstract. We prove the existence of sectors of minimal growth for realizations of boundary value problems on conic manifolds under natural ellipticity conditions. Special attention is devoted to the clarification of the analytic structure of the resolvent. 1.
The convergence of the cascadic conjugategradient method under a deficient regularity
 Problems and Methods in Mathematical Physics (StuttgartLeipzig
, 1994
"... Abstract. We study the convergence properties of the cascadic conjugategradient method (CCGmethod), which can be considered as a multilevel method without coarsegrid correction. Nevertheless, the CCGmethod converges with a rate that is independent of the number of unknowns and the number of grid ..."
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Cited by 7 (2 self)
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Abstract. We study the convergence properties of the cascadic conjugategradient method (CCGmethod), which can be considered as a multilevel method without coarsegrid correction. Nevertheless, the CCGmethod converges with a rate that is independent of the number of unknowns and the number of grid levels. We prove this property for twodimensional elliptic secondorder Dirichlet problems in a polygonal domain with an interior angle greater than π. For piecewise linear finite elements we construct special nested triangulations that satisfy the conditions of a “triangulation of type (h, γ, L)” in the sense of I. Babuˇska, R. B. Kellogg and J. Pitkäranta. In this way we can guarantee both the same order of accuracy in the energy norm of the discrete solution and the same convergence rate of the CCGmethod as in the case of quasiuniform triangulations of a convex polygonal domain. 1.
Fast SemiAnalytic Computation of Elastic Edge Singularities
 Comput. Methods Appl. Mech. Engrg
, 1998
"... The singularities that we consider are the characteristic nonsmooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges. We describe here a method to compute their singularity exponents and the associated angular sin ..."
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Cited by 6 (2 self)
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The singularities that we consider are the characteristic nonsmooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges. We describe here a method to compute their singularity exponents and the associated angular singular functions. We present the implementation of this method in a program whose input data are geometrical data, the elasticity coefficients of each material involved and the type of boundary conditions (Dirichlet, Neumann or mixed conditions). Our method is particularly useful with anisotropic materials and allows to "follow" the dependency of singularity exponents along a curved edge. 1. Introduction In linear elasticity, problems are usually solved with industrial codes using the finite element method. However, when the elastic body has corners on its boundary, such as a polygon or a polyhedron, the solution obtained is inaccurate near the corners. The reason is that on such domains, el...