on the use of perfectly matched layers at corners for scattering problems with sign changing coefficients we investigate in a 2d setting the scattering of time harmonic electromagnetic waves by a plasmonic device represented as a non dissipative bounded and penetrable obstacle with a negative permittivity using the t coercivity approach we first prove that the problem is well posed in the classical framework h loc 1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device when the latter has corners for values inside the critical interval unusual strong singularities for the electromagnetic field can appear in that case well posedness is obtained by imposing a radiation condition at the corners to select the outgoing black hole plasmonic wave that is the one which carries energy towards the corners a simple and systematic criterion is given to define what is the outgoing solution finally we propose an original numerical method based on the use of perfectly matched layers at the corners we emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods

solving maxwell s equations in singular domains with a nitsche type method in this paper we propose and analyze a method derived from a nitsche approach for handling boundary conditions in the maxwell equations several years ago the nitsche method was introduced to impose weakly essential boundary conditions in the scalar laplace operator then it has been worked out more generally and transferred to continuity conditions we propose here an extension to vector div curl problems this allows us to solve the maxwell equations particularly in domains with reentrant corners where the solution can be singular we formulate the method for both the electric and magnetic fields and report some numerical experiments

solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d polyhedral geometries with the help of a continuous approximation of the electromagnetic field in order to remove spurious eigenmodes their method required a parameterization of the variational formulation in order to avoid this difficulty we use a mixed variational setting instead of the parameterization which allows us to handle the divergence free constraint on the field in a straightforward manner the numerical analysis of the method is carried out and numerical examples are provided to show the efficiency of our approach

the fourier singular complement method for the poisson problem part ii axisymmetric domains this paper is the second part of a threefold article aimed at solving numerically the poisson problem in three dimensional prismatic or axisymmetric domains in the first part of this series the fourier singular complement method was introduced and analysed in prismatic domains in this second part the fscm is studied in axisymmetric domains with conical vertices whereas in the third part implementation issues numerical tests and comparisons with other methods are carried out the method is based on a fourier expansion in the direction parallel to the reentrant edges of the domain and on an improved variant of the singular complement method in the 2d section perpendicular to those edges neither refinements near the reentrant edges or vertices of the domain nor cut off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of fourier modes used

a finite element method for a curlcurl graddiv eigenvalue interface problem in this paper we propose and study a finite element method for a curlcurl graddiv eigenvalue interface problem its solution may be of piecewise non h 1 we would like to approximate such a solution in an h 1 conforming finite element space with the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces the proposed method is essentially a standard h 1 conforming element method up to element bubbles which can be statically eliminated at element levels we first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds we then analyze the underlying eigenvalue interface problem and we obtain the error bounds mathcal o h 2r 0 for eigenvalues which correspond to eigenfunctions in prod j 1 j h r omega j 3 hookrightarrow h r 0 omega 3 space where the piecewise regularity r and the global regularity r 0 may belong to the most interesting interval 0 1

numerical solution to the time dependent maxwell equations in axisymmetric singular domains the singular complement method in this paper we present a method to solve numerically the axisymmetric time dependent maxwell equations in a singular domain in math methods appl sci 25 2002 49 math methods appl sci 26 2003 861 the mathematical tools and an in depth study of the problems posed in the meridian half plane were exposed the numerical method and experiments based on this theory are now described here it is also the generalization to axisymmetric problems of the singular complement method that we developed to solve maxwell equations in 2d singular domains see c r acad sci paris t 330 2000 391 it is based on a splitting of the space of solutions in a regular subspace and a singular one derived from the singular solutions of the laplace problem numerical examples are finally given to illustrate our purpose in particular they show how the singular complement method captures the singular part of the solution

continuous galerkin methods for solving the time dependent maxwell equations in 3d geometries a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d geometries with the help of a continuous approximation of the electromagnetic field in this paper we investigate how their framework can be adapted to compute the solution to the time dependent maxwell equations in addition we propose some extensions such as the introduction of a mixed variational setting and its discretization to handle the constraint on the divergence of the field 2007 elsevier inc all rights reserved

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

an asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows it is well known that any viscous fluid flow near a corner consists of infinite series of eddies with decreasing size and intensity unless the angle is larger than a certain critical angle h k moffat j fluid mech 18 1964 pp 1 18 the objective of the current work is to simulate such infinite series of eddies occurring in steady flows in domains with corners the problem is approached by high order finite element method with exponential mesh refinement near the corners coupled with analytical asymptotics of the flow near the corners such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow the method was tested on the problem of the lid driven cavity flow as well as on the problem of the backward facing step flow the results of computations of the lid driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods the results also indicate that the relative error of finding the eddies intensity and position decreases uniformly for all the eddies as the mesh is refined i e the relative error in computation of different eddies does not depend on their size

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

augmented formulations for solving maxwell equations we consider augmented variational formulations for solving the static or time harmonic maxwell equations for that a term is added to the usual h curl conforming formulations it consists of a weighted l2 scalar product between the divergence of the em and the divergence of test fields in this respect the methods we present are h curl div conforming we also build mixed augmented variational formulations with either one or two lagrange multipliers to dualize the equation on the divergence and when applicable the relation on the tangential or normal trace of the field it is proven that one can derive formulations which are equivalent to the original static or time harmonic maxwell equations in the latter case spurious modes are automatically excluded numerical analysis and experiments will be presented in the forthcoming paper augmented formulations for solving maxwell equations numerical analysis and experiments in preparation

a nodal based finite element approximation of the maxwell problem suitable for singular solutions a new mixed finite element approximation of maxwell s problem is proposed its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term which yields a very weak control on the divergence of the unknown the method is shown to be stable and convergent in the natural h rm mathbf curl 0 omega norm for this unknown in particular convergence also applies to singular solutions for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement

a singular field method for maxwell s equations numerical aspects for 2d magnetostatics the present paper deals with the solution of maxwell type problems by means of nodal h1 conforming finite elements in a nonconvex piecewise regular domain surrounded by a perfect conductor such a discretization cannot in general approximate the singular behavior of the electromagnetic field near reentrant corners or edges the singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior the latter are deduced from the singular functions associated with the scalar laplace operator the theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting but the study can be easily extended to numerous maxwell type problems the numerical implementation of both variants is studied for a domain containing a single reentrant corner

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

a combined nodal continuous discontinuous finite element formulation for the maxwell problem abstract continuous galerkin formulations are appealing due to their low computational cost whereas discontinuous galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters since many electromagnetic problems involve materials with different physical properties this last point is very important for this reason in this article we have developed a combined cg dg formulation for maxwell s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches in particular the second formulation we propose comes from a novel continuous galerkin formulation that reduces the amount of stabilization introduced in the numerical system in all cases we have performed stability and convergence analyses of the methods the outcome of this work is a new approach that keeps the low cpu cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones all these methods have been tested using a problem with singular solution and another one with different materials in order to prove that in fact the resulting formulations can properly deal with these problems

phase accuracy comparisons and improved far field estimates for 3 d edge elements on tetrahedral meshes abstract edge element methods have proved very effective for 3 d electromagnetic computations and are widely used on unstructured meshes however the accuracy of standard edge elements can be criticised because of their low order this paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3 d edge element approximations in comparison to other methods scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements and a new formulation of the far field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations

the finite element method for a boundary value problem with strong singularity the existence and uniqueness of the r n generalized solution for the third boundary value problem and the non self adjoint second order elliptic equation with strong singularity are established we construct a finite element method with a basis containing singular functions the rate of convergence of the approximate solution to the r n generalized solution in the norm of the sobolev weighted space is established and finally results of numerical experiments are presented

computation of maxwell singular solution by nodal continuous elements abstract in this paper we propose and analyze a nodal continuous and h 1 conforming finite element method for the numerical computation of maxwell s equations with singular solution in a fractional order sobolev space h r where r may take any value in the most interesting interval 0 1 the key feature of the method is that mass lumping linear finite element l 2 projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of l 2 space we shall use the nodal continuous linear finite elements enriched with one element bubble in each element to approximate the singular and non h 1 solution discontinuous and nonhomogeneous media are allowed in the method some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method

predictor corrector p and hp versions of the finite element method for poisson s equation in polygonal domains abstract we consider boundary value problems for the poisson equation on polygonal domains with general nonhomogeneous mixed boundary conditions and derive on the one hand explicit extraction formulas for the coefficients of the singularities on the other hand the formulas are used to construct efficient adaptations for the h p and h p versions of the finite element method for the numerical treatment a priori error estimates show that the h version of the finite element algorithm exhibits the same rate of convergence as it is known for problems with smooth solutions however the principal results of the present work are the robust exponential convergence results for the p and h p versions of the finite element method on quasiuniform meshes in fact it is shown that if the input data source term and boundary data are piecewise analytic then with appropriate choices of conforming finite element subspaces v n of dimension n n the p and h p versions of the finite element algorithms on quasiuniform meshes yield approximate solutions u n to the exact solution u that satisfy the estimates u u n h 1 c 1 e b 1 n 2 3 and u u n h 1 c 2 e b 2 n 1 3 respectively several numerical experiments are included to illustrate the practical effectiveness of the proposed algorithms the results show that the theoretical error analyses are attained within the range of engineering accuracy

quasi stationary measures for conservative dynamics in the infinite lattice we study quasi stationary measures for conservative particle systems in the in finite lattice existence of quasi stationary measures is established for a fairly general class of reversible systems for the special cases of a system of independent random walks and the symmetric simple exclusion process it is shown that qualitative features of quasi stationary measures change drastically with dimension

time dependent maxwell s equations with charges in singular geometries this paper is devoted to the solution of the instationary maxwell equations with charges the geometry of the domain can be singular in the sense that its boundary can include reentrant corners or edges the difficulties arise from the fact that those geometrical singularities generate in their neighborhood strong electromagnetic fields the time dependency of the divergence of the electric field is addressed to tackle this problem some new theoretical and practical results are presented on curl free singular fields and on singular fields with l2 non vanishing divergence the method which allows to compute the instationary electromagnetic field is based on a splitting of the spaces of solutions into a two term direct sum first the subspace of regular fields it coincides with the whole space of solutions provided that the domain is either convex or with a smooth boundary second a singular subspace defined and characterized via the singularities of the laplace operator several numerical examples are presented to illustrate the mathematical framework this paper is the generalization of the singular complement method

a possibilistic programming approach for closed loop supply chain network design under uncertainty the design of closed loop supply chain networks has attracted more attention in recent years according to business and environmental factors the significance of accounting for uncertainty and risk in such networks spurs an interest to develop appropriate decision making tools to cope with uncertain and imprecise parameters in closed loop supply chain network design problems this paper proposes a bi objective possibilistic mixed integer programming model to deal with such issues the proposed model integrates the network design decisions in both forward and reverse supply chain networks and also incorporates the strategic network design decisions along with tactical material flow ones to avoid the sub optimalities led from separated design in both parts to solve the proposed possibilistic optimization model an interactive fuzzy solution approach is developed by combining a number of efficient solution approaches from the recent literature numerical experiments are conducted to demonstrate the significance and applicability of the developed possibilistic model as well as the usefulness of the proposed solution approach

a singular field method for the solution of maxwell s equations in polyhedral domains it is well known that in the case of a regular domain the solution of the time harmonic maxwell s equations allows a discretization by means of nodal finite elements this is achieved by solving a regularized problem similar to the vector helmholtz equation the present paper deals with the same problem in the case of a nonconvex polyhedron it is shown that a nodal finite element method does not approximate in general the solution to maxwell s equations but actually the solution to a neighboring variational problem involving a different function space indeed the solution to maxwell s equations presents singularities near the edges and corners of the domain that cannot be approximated by lagrange finite elements a new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly this singular field method is presented in various situations such as electri

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

a nonconforming finite element method for a two dimensional curl curl and grad div problem a numerical method for a two dimensional curl curl and grad div problem is studied in this paper it is based on a discretization using weakly continuous p 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries optimal convergence rates up to an arbitrary positive epsilon in both the energy norm and the l 2 norm are established on graded meshes the theoretical results are confirmed by numerical experiments

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

two coupled particle finite volume methods using delaunay voronoi meshes for the approximation of vlasov poisson and vlasov maxwell equations this paper deals with the approximation of vlasov poisson and vlasov maxwell equations we present two coupled particle finite volume methods which use the properties of delaunay voronoi meshes these methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices

numerical approximation of the maxwell equations in inhomogeneous media by ap1conforming finite element method the aim of this paper is to present an extension of thep1conforming finite element method for the time dependent maxwell equations that has been previously exposed we shall consider inhomogeneous media with the piecewise constant dielectric and magnetic parameters and precisely the interface of two such media by analogy with the idea developed we propose a method based on the dualization and then the approximation of the interface conditions in a way consistent with the one derived for the fields and the lagrange multipliers of the divergence constraints inside the domain

a finite element method for domains with corners a new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities in it the singularity is eliminated form the computational domain in an exact fashion this is in contrast to other common methods such as those which use a refined mesh in the singularity region or those which use special singular finite elements in them the singularity is treated as a part of the numerical scheme the new method is illustrated on an elliptic differential equation in a domain containing a re entrant corner numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region both methods require about the same computing time

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically