Martin Costabel

University of Rennes | UR1 · Institut de Recherche Mathématique de Rennes - IRMAR

We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The domains depend on $\epsilon$ only within a ball of radius proportional to $\epsilon$ and, as $\epsilon$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singulari...

Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann–Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular...

For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order $O(h^{2})$ in the interior of the interval and a boundary layer where the consistency error does not tend to zero.

Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann-Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular...

Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function. This characterization is universal in the sense that the equivalence constants are independent of the domain. In particular it...

Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function. This characterization is universal in the sense that the equivalence constants are independent of the domain. In particular it...

This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some systems of coordinates. The boundary conditions are those of the perfectly conducting or perfectly insulating body. The main case of interest is products in Cartesian variables. Cylindrica...

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that the solution of $\Delta u=f$ in $\Omega$ and either $u=0$ on $\partial\Omega$ or $\partial\_{n} u=0$ on $\parti...

This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some system of coordinates. The boundary conditions are those of the perfectly conducting or perfectly insulating body. The main case of interest is products in Cartesian variables. Cylindrical...

Time‐dependent problems that are modeled by initial‐boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation (BIE) method. The ideal situation is when the right‐hand side in the partial differential equation and the initial conditions vanish, the data are given only on the...

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the "functional analytic approach" of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $\epsilon$ for solutions of elliptic boundary value problems. Using the method of layer...

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $\epsilon$ for solutions of elliptic boundary value problems. Using the method of lay...

For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška–Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, is true...

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant...

For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babu{\v s}ka --Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, i...

We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We consider the case of a cylindrical obstacle and fields invariant along the axis of the cylinder, which allows the reduction to two-dimensional problems. With this simplification, we can refine the...

The inf-sup constant for the divergence, or LBB constant, is related to the Cosserat spectrum. It has been known for a long time that on non-smooth domains the Cosserat operator has a non-trivial essential spectrum, which can be used to bound the LBB constant from above. We prove that the essential spectrum on a plane polygon consists of an interva...

Volume integral equations have been used as theoretical and numerical tools in scattering theory for a long time. The basic idea of the VIE method in scattering by a penetrable object is to consider the effect of the scatterer as a perturbation of a whole-space constant coefficient problem and to solve the latter by convolution with the whole-space...

We explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems. The abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors’ paper in [M. Costabel, M. Dauge, S. Nicaise, Analytic regularity for linear ellipt...

In many applications, boundary element methods can only be used in combination with finite elements which cover that part of the domain where inhomogeneities or nonlinearities are located. A lot of detailed descriptions and implementations of such coupling methods exist, see e. g. Brebbia-Telles-Wrobel1 and the references given there.

In this paper we discuss the hp edge finite element approximation of the Maxwell cavity eigenproblem. We address the main arguments for the proof of the discrete compactness property. The proof is based on a conjectured L2 stability estimate for the involved polynomial spaces which has been verified numerically for p≤15 and illustrated with the cor...

In dimension 2, the Horgan-Payne angle serves to construct a lower bound for the inf-sup constant of the divergence arising in the so-called LBB condition. This lower bound is equivalent to an upper bound for the Friedrichs constant. Explicit upper bounds for the latter constant can be found using a polar parametrization of the boundary. Revisiting...

Explicit asymptotic series describing solutions to the Laplace equation in the vicinity of a circular edge in a three-dimensional domain was recently provided in Yosibash et al. (Int J Fract 168:31–52, 2011). Utilizing it, we extend the quasidual function method (QDFM) for extracting the generalized edge flux intensity functions (GEFIFs) along circ...

The equivalence between the inequalities of Babu\v{s}ka-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan-Payne inequality,...

The equivalence between the inequalities of Babuška-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan-Payne inequality, whic...

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch.

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known...

We study the strongly singular volume integral operator that describes the scattering of time-harmonic electromagnetic waves. For the case of piecewise constant material coefficients and smooth interfaces, we determine the essential spectrum. We show that it is a finite set and that the operator is Fredholm of index zero in H(curl)H(curl) if and on...

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without lo...

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary inte...

Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (...

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface–volume system of integral equations with weakly singular kernels. The analysis of the coupled system is bas...

Suppose that $\Omega$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T $ which preserves support in $\bar{\Omega$}, is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that $dT$ is the identity...

This is a preliminary version of the first part of a book project that will consist of four parts. We are making it available in electronic form now, because there is a demand for some of the technical tools it provides, in particular a detailed presentation of analytic elliptic regularity estimates in the neighborhood of smooth boundary points. In...

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary inte...

We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovski˘ ı's integral operator, acting on differential forms in R n. We prove that these operators are pseudodifferential operators of order −1. The Poincaré-type operators map polynomials to polynomials and can h...

dans : Computational Electromagnetism and Acoustics Organised by Ralf Hiptmair, Zürich; Ronald H. W. Hoppe, Augsburg; Patrick Joly, Le Chesnay; Ulrich Langer, Linz February 14th - February 20th, 2010

On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf.\ papers by Kondrat'ev and Maz'ya-Plamenevskii. Two classes of weighted norms are usually considered: Homogeneous norms, wh...

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface $\Gamma$. In this paper, following an idea developed by Kleinman and Martin \cite{KlMa} for acoustic scatter...

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface $\Gamma$. In this paper, following an idea developed by Kleinman and Martin \cite{KlMa} for acoustic scatter...

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in th...

This paper is devoted to the construction of continuous trace lifting operators compatible with the de Rham complex on the reference hexahedral element (the unit cube). We consider three trace operators: The standard one from H^1 , the tangential trace from boldsymbol{mathit{H}}(mathbf{curl}) and the normal trace from boldsymbol{mathit{H}}(mathrm...

We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiĭ’s integral operator, acting on differential forms in _boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose^n{\mathbb{R}^n} . We prove that these operators are pseudodifferential opera...

The workshop Analysis of Boundary Element Methods , organized by Martin Costabel (Rennes) and Ernst P. Stephan (Hannover). This meeting brought together 46 experts in numerical analysis. Boundary element methods (BEM) are well established numerical methods with a wide range of applications. There are still many challenging problems, the aim of this...

Report No. 19/2008 Analysis of Boundary Element Methods Organised by Martin Costabel, Rennes; Ernst P. Stephan, Hannover April 13th - April 19th, 2008

Variational arguments go back a long time in the history of boundary integral equations. Energy methods have shown up very early, then virtually disappeared from the common knowledge and eventually resurfaced in the context of boundary element methods. We focus on some not so well known parts of classical works by well known classical authors and d...

This paper is devoted to the construction of continuous trace lift- ing operators compatible with the de Rham complex on the reference hexahe- dral element (the unit cube). We consider three trace operators: The standard one from H1, the tangential trace from H(curl) and the normal trace from H(div). For each of them we construct a continuous right...

We consider a quasistatic system involving a Volterra kernel modelling an hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possibly anisotropic material law. We study the a...

We consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness ε and with external boundary conditions of Dirichlet or Neumann type. Our aim is to build a multi-scale expansion as ε goes to 0 for that solution. After presenting a complete multi-scale expansion in a smooth coated domain, we focus on the c...

Discretization of Maxwell eigenvalue problems with edge finite elements involves a simultaneous use of two discrete subspaces of H^1 and H(rot), reproducing the exact sequence condition. Kikuchi's Discrete Compactness Property, along with appropriate approximability conditions, implies the convergence of discrete eigenpairs to the exact ones. In th...

The solution to elastic isotropic problems in three-dimensional (3-D) polyhedral domains has an explicit structure in the vicinity of the edges. This structure involves a family of eigen-functions with their shadows, and the associated Edge Stress Intensity Functions (ESIFs), which are functions along the edges. For the extraction of ESIFs, we appl...

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regu- larization by a divergence term is a standard tool to obtain e quivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization con- verge to wrong solutions in any non-convex polygon. Modifica ti...

We study approximation errors for the h-version of Ndlec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error est...

Time‐dependent problems that are modeled by initial‐boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation method. The ideal situation is when the right‐hand side in the partial differential equation and the initial conditions vanish, the data are given only on the bound...

The asymptotics of solutions to scalar second order elliptic boundary value problems in three-dimensional polyhedral domains in the vicinity of an edge is provided in an explicit form. It involves a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the edges. Utilizing...

The aim of this work is to provide a description of the corner asymp-totics for the solutions of Maxwell equations in and outside a conductor body and to investigate the limit as the ratio permittivity/conductivity tends to zero (the eddy current limit). Corner singularities of the Maxwell transmission problem and also of the eddy current model hav...

We present a method for the computation of the coefficients of singularities along the edges of a polyhedron for second order elliptic boundary value problems. The class of problems considered includes problems of stress concentration along edges or crack fronts in general linear three-dimensional elasticity. Our method uses an incom- plete constru...

We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit...

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, wher...

We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface M with boundary E . The same boundary conditions are prescribed on both sides of the surface M . The most important model behind this investigation is the crack problem in three-dimensional linear elasticity (either isotropic or anisotropic): ther...

We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane....

We address the computation by finite elements of the non-zero eigenvalues of the (curl, curl) bilinear form with perfect conductor boundary conditions in a polyhedral cavity. One encounters two main difficulties: (i) The infinite dimensional kernel of this bilinear form (the gradient fields), (ii) The unbounded singularities of the eigen-fields nea...

In the presence of re-entrant comers or edges, the standard regularization of Maxwell equations combined with the use of nodal finite elements is known to produce wrong solutions. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. We present numerical experiments consisting in the computatio...

The electric field integral equation on an open surface is transformed into a strongly elliptic system using Hodge decomposition on the surface. The resulting system of pseudodifferential equations is discretized by finite elements using nodal-based wavelet bases. The necessary function spaces are described and results about matrix compression, sta...

We study tangential vector fields on the boundary of a bounded Lipschitz domain Ω in ℝ 3. Our attention is focused on the definition of suitable Hilbert spaces corresponding to fractional Sobolev regularities and also on the construction of tangential differential operators on the non-smooth manifold. The theory is applied to the characterization o...

We present a new method of regularizing time harmonic Maxwell equations by a grad-div term 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...

Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On...

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ℝn+1×ℝn+1, and then develop a periodisation procedure for the calculus of symbols on the cylinder\(\mathbb{T}^n \)×ℝ. We show Gårding's inequality for suitable operators and p...

The singularities that we consider are the characteristic non-smooth 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...

. We consider general homogeneous Agmon-Douglis-Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two-dimensional domain. We prove that the singular functions expressed in polar coordinates (r; ) near the crack tip all have the form r 1 2 +k '() with k 0 integer,...

It is well known that, in the presence of non-convex corners or edges on the boundary, nodal finite elements associated with a conformal curl-div formulation do not converge to the correct limit when the electric or magnetic boundary conditions are also imposed in the discrete space. The authors formulate and investigate in a simple two-dimensional...

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 sing...

. We consider spline collocation methods for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover for example the case of the single layer heat operator equation when the spatial domain is a disc. 1. Introduction. The integral equation method f...

. In a convex polyhedron, a part of the Lam'e eigenvalues with hard simple support boundary conditions does not depend on the Lam'e coefficients and coincide with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lam'e coefficients and the associated eigenmodes are the gradients of the Laplace-Dirichlet e...

We show the density of smooth functions in the space of square integrable vector fields whose curl, divergence, and tangential (or normal) trace on the boundary are square integrable. Our proof is based on the H3/2 regularity for the Dirichlet and Neumann problems and constitutes a simplification and generalization of [3].

We show the density of smooth functions in the space of square integrable vector fields whose curl, divergence, and tangential (or normal) trace on the boundary are square integrable. Our proof is based on the H3/2 regularity for the Dirichlet and Neumann problems and constitutes a simplification and generalization of [3].

. As a simplified model for contact problems, we study a mixed Neumann-Robin boundary value problem for the Laplace operator in a smooth domain in R 2 . The Robin condition contains a small parameter " inducing boundary layers of corner type at the transition points as proved in [4]. We present an integral equation for the numerical solution of thi...

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 non-convex edges, the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the L...

this article, we will try to give a survey over some of the main mathematical ideas involved. Given the fast growth of the field of BEM, both on the side of engineering applications and on the side of mathematical and numerical analysis, the list of references cannot, of course, be comprehensive. 2 Some highlights of the mathematical history of tim...

We provide a description of the non regular parts of the fields involved in Maxwell equations, first at the variational level, then at the solution level when the right hand side has regularity properties. (For the ideas of the proofs, see [5]. Details will be published elsewhere.)

We provide a description of the non regular parts of the fields involved in Maxwell equations, first at the variational level, then at the solution level when the right hand side has regularity properties. (For the ideas of the proofs, see [5]. Details will be published elsewhere.).

We prove the density of regular fielclx in the space of square-integrahle fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electmmagnetism equations.

We prove the density of regular fields in the space of square-integrable fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electromagnetism equations.

We study the unique solvability of a 2 × 2 system of boundary integral equations arising from a single layer potential representation for the bihar- monic Dirichlet problem. We answer the question "for which curves does this system have a unique solution ?" as follows: For each curve there are between 1 and 4 exceptional scale factors ρ such that o...

We study a mixed Neumann-Robin boundary value problem for the Laplace operator in a smooth domain in R^2. The Robin condition contains a parameter ε and tends to a Dirichlet condition as ε tends to zero. We give a complete asymptotic expansion of the solution in powers of ε. At the points where the boundary conditions change, there appear boundary...

Solutions of linear elliptic partial differential equations in unbounded domains can be represented by boundary potentials if they satisfy certain conditions at infinity. These radiation conditions depend on the fundamental solution chosen for the integral representation. We prove some basic results about radiation conditions in a rather general fr...

The 2×2 system of integral equations corresponding to the biharmonic single layer potential in ℝ2 is known to be strongly elliptic. It is also known to be positive definite on a space of functions orthogonal to polynomials of degree one. We study the question of its unique solvability without this orthogonality condition. To each curve Γ, we associ...

We consider boundary value problems for elliptic systems in the sense of Agmon-Douglis-Nirenberg on plane domains with corners, where the domain, the coefficients of the operators and the right hand sides all depend on a parameter. We construct corner singularities in such a way that the corresponding decomposition of the solution into regular and...

This paper is concerned with approximation methods for Neumanns integral equation on curves with corners. Necessary and sufficient conditions for the stability of the piecewise constant c-collocation and for the quadrature method, using the rectangular rule, are given.

We study the numerical approximation of two pseudodifferential equations on a plane rectangle defined by the single layer potential and by the normal derivative of the double layer potential of the three-dimensional Laplacian. The solutions are approximated by nodal collocation with piecewise bilinear, respectively by bicubic, trial functions on a...