a locally divergence free interior penalty method for two dimensional curl curl problems an interior penalty method for certain two dimensional curl curl problems is investigated in this paper this method computes the divergence free part of the solution using locally divergence free discontinuous p 1 vector fields on graded meshes it has optimal order convergence up to an arbitrarily small epsilon for the source problem and the eigenproblem results of numerical experiments that corroborate the theoretical results are also presented

c 0 elements for generalized indefinite maxwell equations in this paper we develop the c 0 finite element method for a generalized curlcurl grad div indefinite maxwell problem in a lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the h r regularity for some r 1 the ingredients of our method are that two mass lumping l 2 projectors are applied to curl and div operators in the problem and that c 0 linear element or isoparametric bilinear element enriched with one element bubble on every triangle element or with two element bubbles on every quadrilateral element respectively is employed for each component of the nonsmooth solution due to the fact that the element bubbles can be statically eliminated at element levels our method is essentially three nodes or four nodes c 0 lagrange element method with two fortin type interpolations established a very technical duality argument is elaborated to estimate the error for the indefinite problem for the nonsmooth solution having the h r regularity where r may vary in the interval 0 1 we obtain the error bound mathcal o h r in an energy norm some numerical experiments are performed to confirm the theoretical error bounds

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 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

graph rules for the linked cluster expansion of the legendre effective action graph rules for the linked cluster expansion of the legendre effective action are derived and proven in d 2 euclidean dimensions a key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs the hopping interaction is allowed to be long ranged and scale dependent thereby producing an in principle exact solution of s functional renormalization group equation graph rules for the linked cluster expansion of the legendre effective action are derived and proven in d 2 euclidean dimensions a key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs the hopping interaction is allowed to be long ranged and scale dependent thereby producing an in principle exact solution of s functional renormalization group equation

a high order accurate parallel solver for maxwell s equations on overlapping grids a scheme for the solution of the time dependent maxwell s equations on composite overlapping grids is described the method uses high order accurate approximations in space and time for maxwell s equations written as a second order vector wave equation high order accurate symmetric difference approximations to the generalized laplace operator are constructed for curvilinear component grids the modified equation approach is used to develop high order accurate approximations that use only three time levels and have the same time stepping restriction as the second order scheme discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed the implementation is optimized for component grids that are cartesian resulting in a fast and efficient method the solver runs on parallel machines with each component grid distributed across one or more processors numerical results in two and three dimensions are presented for the fourth order accurate version of the method these results demonstrate the accuracy and efficiency of the approach

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

second order analysis of control affine problems with scalar state constraint in this article we establish new second order necessary and sufficient optimality conditions for a class of control affine problems with a scalar control and a scalar state constraint these optimality conditions extend to the constrained state framework the goh transform which is the classical tool for obtaining an extension of the legendre condition

a discontinuous galerkin method on refined meshes for the two dimensional time harmonic maxwell equations in composite materials in this paper a discontinuous galerkin method for the two dimensional time harmonic maxwell equations in composite materials is presented the divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized due to an appropriate mesh refinement near exterior and interior corners the singular behaviour of the electromagnetic field is taken into account optimal error estimates in a discrete energy norm and in the l 2 norm are proved in the case where the exact solution is singular

singularities of electromagnetic fields in polyhedral domains 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 laplace operator 4 5 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 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 10

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

a model for cancer chemotherapy with state space constraints abstract the analysis of optimal control problems is significantly more difficult when state space constraints are imposed and thus in many modelling approaches explicit constraints are handled by including penalty terms in the objective in this paper we use the example of a simple model for cancer chemotherapy to compare the structure of extremals when upper limits on the total number of cancer cells are imposed a implicitly through a penalty term in the objective and b explicitly in the form of state space constraints

mayer expansions and the hamilton jacobi equation we review the derivation of wilson s differential equation in infinitely many variables which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed we show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic the resulting series solutions are generalizations of the mayer expansion in statistical mechanics the differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated mayer expansions applications include the yukawa gas in two dimensions past the 4 threshold and another derivation of some earlier results of gopfert and mack

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

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

analysis of a continuous finite element method for h mathrm curl mathrm div elliptic interface problem in this paper we develop a continuous finite element method for the curlcurl grad div vector second order elliptic problem in a three dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials in spite of the fact that the curlcurl grad div interface problem is closely related to the elliptic interface problem of vector laplace operator type the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges to discretize the curlcurl grad div interface problem by the continuous finite element method we apply an element local l 2 projector to the curl operator and a pseudo local l 2 projector to the div operator where the continuous lagrange linear element enriched by suitable element and face bubbles may be employed it is shown that the finite element problem retains the same coercivity property as the continuous problem an error estimate mathcal o h r in an energy norm is obtained between the analytical solution and the continuous finite element solution where the analytical solution is in prod l 1 l h r omega l 3 for some r in 1 2 1 due to the domain boundary reentrant corners and edges e g nonconvex polyhedron and due to the interfaces between the different material domains in omega bigcup l 1 l omega l

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

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

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 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 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

a new and simple implementation of the element local l2 projected continuous finite element method in this paper we present a new and simple method modified from the original element local l 2 projected continuous finite element method to resolve some static electromagnetic problems in two dimension this numerical method is established on the least square process to minimize the total error s energy where the residual due to an additional element local projection is considered many numerical experiments are presented for the electrostatic problem the maxwell s equations and the maxwell s eigenvalue problem and the numerical performances are excellent with the optimal convergence orders even for the non h 1 solution

iterative solutions for electromagnetic fields at perfectly reflective and transmissive interfaces using clifford algebra and the multidimensional cauchy integral a technique is presented for calculating the solution of maxwell s equations using a cfie based on the cauchy integral and formulated in the guise of clifford algebra this formulation has a geometric interpretation leading to an iterative method of solution which is easily proven as convergent and correct for both perfectly reflective and perfectly transmissive interfaces simple test cases involving a cubic boundary and plane wave and dipole sources are used to investigate the numerical performance of the method