recovery of a time dependent source from a surface measurement in maxwell s equations we study an inverse source problem arising in computations of electromagnetic fields governed by maxwell s equations the unknown source is solely time dependent and it will be recovered from a surface measurement the existence and uniqueness of a solution to this inverse problem is shown moreover a constructive algorithm for approximations is designed and its convergence is established

a mixed formulation for the brinkman problem the brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium where fractures bubbles or channels alternate inside a porous matrix in this work we explore a novel mixed formulation of the brinkman problem based on the hodge decomposition of the vector laplacian introducing the flow s vorticity as an additional unknown this formulation allows for a uniformly stable and conforming discretization by standard finite elements nedelec raviart thomas piecewise discontinuous a priori error estimates for the discretization error in the h rm curl omega h rm div omega l 2 omega norm of the solution which are optimal with respect to the approximation properties of finite element spaces are obtained the theoretical results are illustrated with numerical experiments finally the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for h rm curl and h

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

formulation of the eddy current problem in multiply connected regions in terms of h in this paper various formulations for the eddy current problem are presented the formulations are based on solving directly for the magnetic field h and they differ from each other mainly by how the field on the boundary is treated the electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region this provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains besides the fivefold decomposition another essential tool in our approach is whitney elements as they provide the structure needed to retain consistency between the continuous and discrete problems the paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems 1998 john wiley sons ltd

hybrid electric magnetic methods in eddy current problems we discuss a method of approximation of eddy current equations in bounded space that relies on the use of the electric field as primary variable in the conductive parts and of the magnetic field in the air theoretical issues about this hybrid approach are addressed is the problem well posed and one shows that edge elements offer a straightforward way to discretize such formulations the now popular a v method is then shown to come as a variant of this formulation in such a way that it allows the use of standard nodal elements however attention is called on the fact that the numerical convergence of the latter method may be in question

a mortar edge element method with nearly optimal convergence for three dimensional maxwell s equations in this paper we are concerned with mortar edge element methods for solving three dimensional maxwell s equations a new type of lagrange multiplier space is introduced to impose the weak continuity of the tangential components of the edge element solutions across the interfaces between neighboring subdomains the mortar edge element method is shown to have nearly optimal convergence under some natural regularity assumptions when nested triangulations are assumed on the interfaces a generalized edge element interpolation is introduced which plays a crucial role in establishing the nearly optimal convergence the theoretically predicted convergence is confirmed by numerical experiments

the hp local discontinuous galerkin method for low frequency time harmonic maxwell equations the local discontinuous galerkin method for the numerical approximation of the time harmonic maxwell equations in a low frequency regime is introduced and analyzed topologically nontrivial domains and heterogeneous media are considered containing both conducting and insulating materials the presented method involves discontinuous galerkin discretizations of the curl curl and grad div operators derived by introducing suitable auxiliary variables and so called numerical fluxes an hp analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained

computation of stationary 3d halo currents in fusion devices with accuracy control this paper addresses the calculation of the resistive distribution of halo currents in three dimensional structures of large magnetic confinement fusion machines a neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error an irrotational electric field is obtained by a geometric formulation based on the electric scalar potential whereas three geometric formulations are compared to obtain a solenoidal current density a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed hybrid finite elements the electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre processing it requires to solve this challenging problem we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state of the art solutions proposed in literature believing that our results are useful in other fields of scientific computing the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented

global well posedness and blow up of solutions for the camassa holm equations with fractional dissipation consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power gamma ge 0 on the well posedness of the camassa holm equations with fractional dissipation it is shown that the zero filter limit alpha rightarrow 0 of the camassa holm equation with fractional dissipation is the fractal burgers equation it is known that in the supercritical case gamma in 0 1 the fractal burgers equation blows up in finite time in h s mathbb r with s frac 3 2 gamma it is established here that the dissipative camassa holm equation is globally well posed in the critical sobolev space h frac 3 2 gamma mathbb r with the fractional parameter gamma in frac 1 2 1 moreover it is also demonstrated that the solution of the dissipative camassa holm equation blows up in finite time in the particular supercritical case when gamma 0

a finite element method with lagrange multipliers for low frequency harmonic maxwell equations the aim of this paper is to analyze a finite element method to solve the low frequency harmonic maxwell equations in a bounded domain containing conductors and dielectrics this system of partial differential equations is a model for the so called eddy currents problem after writing this problem in terms of the magnetic field it is discretized by nedelec edge finite elements on a tetrahedral mesh error estimates are easily obtained if the curl free condition is imposed on the elements in the dielectric domain then the curl free condition is imposed at a discrete level by introducing a piecewise linear multivalued potential the resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential moreover this approach leads to an important saving in computational effort problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account implementation issues are discussed including an amenable procedure to impose the boundary conditions by means of a lagrange multiplier finally the method is applied to solve a three dimensional model problem a cylindrical electrode surrounded by dielectric

numerical solution of eddy current problems in bounded domains using realistic boundary conditions abstract the aim of this paper is to analyze a finite element method to solve the low frequency harmonic maxwell equations in a bounded domain containing conductors and dielectrics and using realistic boundary conditions in that they can be easily measured these equations provide a model for the so called eddy currents the problem is formulated in terms of the magnetic field this formulation is discretized by using nedelec edge finite elements on a tetrahedral mesh error estimates are easily obtained when the curl free condition is imposed explicitly on the elements in the dielectric domain a multivalued magnetic scalar potential is introduced then to impose this curl free condition the discrete counterpart of this formulation leads to an important saving in computational effort problems related to the topology are also considered more precisely the possibility of having a non simply connected dielectric domain is taken into account finally the method is applied to solve two three dimensional model problems a test with a known analytical solution and the computation of the electromagnetic field in a metallurgical arc furnace

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

a vorticity based fully mixed formulation for the 3d brinkman darcy problem abstract we propose and analyze a fully mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium an array of low concentration fixed particles described by brinkman equations and its interaction with non viscous flow within classical porous media governed by darcy s law the system is formulated in terms of velocity and pressure in the porous medium together with vorticity velocity and pressure of the viscous fluid in addition and for sake of the analysis the tangential component of the vorticity is supposed to vanish on the whole boundary of the brinkman domain whereas null normal components of both velocities are assumed on the respective boundaries except on the interface where suitable transmission conditions are considered in this way the derivation of the corresponding mixed variational formulation leads to a lagrange multiplier enforcing the pressure continuity across the interface whereas mass balance results from essential boundary conditions on each domain as a consequence a typical saddle point operator equation is obtained and hence the classical babuska brezzi theory is applied to establish the well posedness of the continuous and discrete schemes in particular we remark that the continuous and discrete inf sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema which constitutes a previously known technique recently denominated t coercivity in turn and consistently with the above the stability of the galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives which yields raviart thomas and nedelec finite element subspaces as feasible choices then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the brinkman momentum equation finally several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

representation theory in complex rank i p deligne defined interpolations of the tensor category of representations of the symmetric group s n to complex values of n namely he defined tensor categories rep s t for any complex t this construction was generalized by f knop to the case of wreath products of s n with a finite group generalizing these results we propose a method of interpolating representation categories of various algebras containing s n such as degenerate affine hecke algebras symplectic reflection algebras rational cherednik algebras etc to complex values of n we also define the group algebra of s n for complex n study its properties and propose a schur weyl duality for rep s t

magnetostatic field computations in terms of two component vector potentials in this paper the magnetostatic problem is stated in terms of two component electric and magnetic vector potentials an associated numerical method based on the adoption of edge elements is proposed this procedure overcomes the cancellation problems and the complexity of the interface conditions encountered by similar approaches in the presence of magnetic inhomogeneities and discontinuities of currents and magnetic fields

volume integral equations in non linear 3 d magnetostatics in this paper a discussion of volume integral formulations in three dimensional non linear magnetostatics is presented integral formulations are examined in connection with whitney s elements in order to find new approaches a numerical algorithm based on a formulation implying properly the continuity conditions of magnetic field strength h i e an h type formulation is introduced results of demanding application problems are shown demonstrating the characteristics of this kind of volume integral approach in addition a discussion of the parallelized version of the numerical code based on the h type approach is presented appended with numerical results illustrating the advantages of combining integral formulations with concurrent computing

an iterative substructuring method for raviart thomas vector fields in three dimensions the iterative substructuring methods also known as schur complement methods form one of two important families of domain decomposition algorithms they are based on a partitioning of a given region on which the partial differential equation is defined into nonoverlapping substructures the preconditioners of these conjugate gradient methods are then given in terms of local problems defined on individual substructures and pairs of substructures and in addition a global problem of low dimension an iterative method of this kind is introduced for the lowest order raviart thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure the theoretical bounds are confirmed by a series of numerical experiments

a formulation of the eddy current problem in the presence of electric ports the time harmonic eddy current problem with either voltage or current intensity excitation is considered we propose and analyze a new finite element approximation of the problem based on a weak formulation where the main unknowns are the electric field in the conductor a scalar magnetic potential in the insulator and for the voltage excitation problem the current intensity the finite element approximation uses edge elements for the electric field and nodal elements for the scalar magnetic potential and an optimal error estimate is proved some numerical results illustrating the performance of the method are also presented

mixed finite element methods for stationary incompressible magneto hydrodynamics a new mixed variational formulation of the equations of stationary incompressible magneto hydrodynamics is introduced and analyzed the formulation is based on curl conforming sobolev spaces for the magnetic variables and is shown to be well posed in possibly non convex lipschitz polyhedra a finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf sup stable velocity pressure space pairs and the magnetic ones by a mixed approach using nedelec s elements of the first kind an error analysis is carried out that shows that the proposed finite element approximation leads to quasi optimal error bounds in the mesh size

on the approximation of electromagnetic fields by edge finite elements part 1 we propose sharp results on the numerical approximation of low regularity electromagnetic fields by edge finite elements we consider general geometrical settings including topologically non trivial domains or domains with a non connected boundary in the model the electric permittivity and magnetic permeability are symmetric tensor valued piecewise smooth coefficients in all cases the error can be bounded by h times a constant where h is the meshsize for some exponent 0 1 that depends both on the geometry and on the coefficients it relies either on classical estimates when 1 2 or on a new combined interpolation operator when 1 2 the optimality of the value of is discussed with respect to abstract shift theorems in some simple configurations typically for scalar valued permittivity and permeability the value of can be further characterized this paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements

critical analysis of the spanning tree techniques two algorithms based upon a tree cotree decomposition called in this paper spanning tree technique stt and generalized spanning tree technique gstt have been shown to be useful in computational electromagnetics the aim of this paper is to give a rigorous description of the gstt in terms of homology and cohomology theories together with an analysis of its termination in particular the authors aim to show by concrete counterexamples that various problems related with both stt and gstt algorithms exist the counterexamples clearly demonstrate that the failure of stt and gstt is not an exceptional event but something that routinely occurs in practical applications

numerical analysis of electric field formulations of the eddy current model this paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current subjected to boundary conditions involving only data easily available in applications two different cases are considered depending on the boundary data input current intensities or differences of potential weak formulations in terms of the electric field are given in both cases in the first one the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit in the second one the electric potentials are imposed by means of lagrange multipliers which are proved to represent the input current intensities optimal error estimates are proved in both cases and implementation issues are discussed finally numerical tests confirming the theoretical results are reported

coupling of viscous and inviscid stokes equations via a domain decomposition method for finite elements a generalized stokes problem is addressed in the framework of a domain decomposition method in which the physical computational domain is partitioned into two subdomains 1 and 2 three different situations are covered in the former the viscous terms are kept in both subdomains then we consider the case in which viscosity is dropped out everywhere in finally a hybrid situation in which viscosity is dropped out only in 1 is addressed the latter is motivated by physical applications in all cases correct transmission conditions across the interface between 1 and 2 are devised and an iterative procedure involving the successive resolution of two subproblems is proposed the numerical discretization is based upon appropriate finite elements and stability and convergence analysis is carried out we also prove that the iteration by subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

construction of a finite element basis of the first de rham cohomology group and numerical solution of 3d magnetostatic problems we devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three dimensional magnetostatic problems in particular we construct a finite element basis of the first de rham cohomology group of the computational domain the proposed method works for general topological configurations and does not need the determination of cutting surfaces

a nonoverlapping domain decomposition method for maxwell s equations in three dimensions in this paper we propose a nonoverlapping domain decomposition method for solving the three dimensional maxwell equations based on the edge element discretization for the schur complement system on the interface we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping subdomains it is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients

stabilized interior penalty methods for the time harmonic maxwell equations we propose stabilized interior penalty discontinuous galerkin methods for the indefinite time harmonic maxwell system the methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field we prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach

simplicial finite elements for scattering problems in electromagnetism abstract scattering problems in electromagnetism imply the approximate computation of vector valued fields to do this with finite elements which were originally designed for scalar fields that is nodal elements is not correct for such vectorial finite elements have serious drawbacks they impose on fields which they are meant to approximate electric field e and magnetic field h a continuity of all components across inter element boundaries which is not a necessary property of the fields simplicial finite elements and more especially edge elements a brand of vector finite elements with degrees of freedom associated with the edges of the mesh are as shown here free of such drawbacks first used in connection with eddy current problems they are shown to fit the scattering problem just as well with no penalty in terms of computational time and with the expected additional advantage of not generating spurious modes when applied to the problem of the eigenmodes of resonating cavities

on the decomposition of the tensor algebra of the classical lie algebras abstract let g n be the lie algebra gln c let s g n be the symmetric algebra of g n and let t g n be the tensor algebra of g n in a recent paper r k gupta studied certain sequences of representations r rn n 1 where rn is a representation of g n these sequences have the property that every irreducible representation occurring in s g n is in exactly one of these sequences fixing f she considers s r f which is the limit on n of the multiplicity of rn in sf g n the fth graded piece of s g n she and r p stanley independently showed that the limit s r f exists and is given by an amazingly elegant formula they call s r f the stable multiplicity of rn in sf g n in this paper an entirely different approach is used to extend the above result in several directions appropriately defined sequences r for all of the classical lie algebras g n are studied and a simple formula for the stable multiplicity m r f g of rn in the isotypic component of tf g n where is any irreducible character of the symmetric group tsf is obtained as in the work of gupta and stanley the expressions for m r f g are amazingly simple special cases include the stable decomposition of the tensor algebra the symmetric algebra and the exterior algebra of g n as a byproduct of our proof a stable decomposition of every isotypic component of t g n is obtained this combinatorial decomposition is in some sense a generalization of kostant s decomposition of s g n into direct sum of the harmonics and the ideal generated by the invariants of positive degree to be precise for f g n projects onto kostant s decomposition of sf g n

substructuring preconditioners for saddle point problems arising from maxwell s equations in three dimensions this paper is concerned with the saddle point problems arising from edge element discretizations of maxwell s equations in a general three dimensional nonconvex polyhedral domain a new augmented technique is first introduced to transform the problems into equivalent augmented saddle point systems so that they can be solved by some existing preconditioned iterative methods then some substructuring preconditioners are proposed with very simple coarse solvers for the augmented saddle point systems with the preconditioners the condition numbers of the preconditioned systems are nearly optimal namely they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size

error analysis of a finite element integral equation scheme for approximating the time harmonic maxwell system in 1996 hazard and lenoir suggested a variational formulation of maxwell s equations using an overlapping integral equation and volume representation of the solution siam j math anal 27 1996 pp 1597 1630 they suggested a numerical scheme based on this approach but no error analysis was provided in this paper we provide a convergence analysis of an edge finite element scheme for the method the analysis uses the theory of collectively compact operators its novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation

error estimates for the full discretization of a nonlocal parabolic model for type i superconductors a vectorial nonlocal linear parabolic problem in terms of the magnetic field for superconductors of type i is considered this problem is obtained from the quasi static maxwell equations the two fluid model of london and london and the nonlocal representation of the superconductive current by eringen space convolution in this contribution a linear fully discrete approximation scheme is proposed to solve this problem the convergence of the scheme is proved and the corresponding error estimates are derived under appropriate assumptions it is also shown how to improve the error estimates under higher regularity

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

analysis of finite element approximation for time dependent maxwell problems we provide an error analysis of finite element methods for solving time dependent maxwell problem using nedelec and thomas raviart elements we study the regularity of the solution and develop some new error estimates of nedelec finite elements as a result the optimal l 2 error bound for the semidiscrete scheme is obtained

bases for coordinate rings of conjugacy classes of nilpotent matrices abstract a general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix this conjecture is proven when the partition given by the transpose jordan type of the nilpotent matrix is a hook or has two parts

analysis of a fem bem model posed on the conducting domain for the time dependent eddy current problem the three dimensional eddy current time dependent problem is considered we formulate it in terms of two variables one lying only on the conducting domain and the other on its boundary we combine finite elements fem and boundary elements bem to obtain a fem bem coupled variational formulation we establish the existence and uniqueness of the solution in the continuous and the fully discrete case finally we investigate the convergence order of the fully discrete scheme

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

analysis of a stabilized finite element method for stokes equations of velocity boundary condition and of pressure boundary condition abstract in this paper a new pressure stabilized finite element method is analyzed for solving the stokes equations the key feature of the method is using the curl integral instead of the standard dirichlet integral and using the pressure stabilization instead of the inf sup condition thus the method is very flexibly applicable to two problems of stokes equations in terms of velocity and pressure with either pressure dirichlet boundary condition or velocity dirichlet boundary condition for both stokes problems the finite element scheme and the finite element space of velocity are the same and one where only the finite element spaces of pressure have small differences a general analysis of stability and error estimates is developed and applications to both stokes problems are further analyzed the method covers the low regularity solution of very weak non h 1 space solution of stokes problem of pressure dirichlet boundary condition as well as higher regularity solutions of stokes problems of either pressure dirichlet boundary condition or velocity dirichlet boundary condition numerical results are presented to illustrate the performance of the method and to confirm the theoretical results

numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via lagrange multipliers this paper deals with the finite element solution of the eddy current problem in a bounded conducting domain crossed by an electric current and subject to boundary conditions appropriate from a physical point of view two different cases are considered depending on the boundary data input current density flux or input current intensities the analysis of the former is an intermediate step for the latter which is more realistic in actual applications weak formulations in terms of the magnetic field are studied the boundary conditions being imposed by means of appropriate lagrange multipliers the resulting mixed formulations are analyzed depending on the regularity of the boundary data finite element methods are introduced in each case and error estimates are proved finally some numerical results to assess the effectiveness of the methods are reported

a finite element method for the magnetostatic problem in terms of scalar potentials the aim of this paper is to analyze a numerical method for solving the magnetostatic problem in a three dimensional bounded domain containing prescribed currents and magnetic materials the method discretizes a well known formulation of this problem based on two scalar potentials the total potential defined in magnetic materials and the reduced potential defined in dielectric media and in nonmagnetic conductors carrying currents the topology of the domain of each material is not assumed to be trivial the resulting variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements transmission conditions are imposed by means of a piecewise linear lagrange multiplier on the surface separating the domains of both potentials error estimates for the numerical method are proved and the results of some numerical tests are reported to assess the performance of the method

finite element simulation of eddy current problems using magnetic scalar potentials we propose a new implementation of the finite element approximation of eddy current problems using as the principal unknown the magnetic field in the non conducting region a scalar magnetic potential is introduced the method can deal automatically with any topological configuration of the conducting region and being based on the search of a scalar magnetic potential in the non conducting region has the advantage of making use of a reduced number of unknowns several numerical tests are presented for illustrating the performance of the proposed method in particular the numerical simulation of a new type of transformer of complicated topological shape is shown

a mixed method for approximating maxwell s equations a semidiscrete mixed finite element approximation to the time dependent maxwell s equations on a bounded smooth domain is analyzed a variational problem for the electric and magnetic fields in which the boundary conditions are enforced naturally is derived then a general convergence result for mixed methods is proven and it is shown how this result may be used to prove various error estimates when nedelec s curl conforming finite elements are used jz3 with connected boundary f and unit outward normal n let e x and 1 x denote respectively the dielectric constant and magnetic permeability of the medium occupying q let a x denote the conductivity of the medium then if e x t and h x t denote respectively the electric and magnetic fields maxwell s equations 10 state that

two level additive preconditioners for edge element discretizations of time harmonic maxwell equations two level additive preconditioners are presented for edge element discretizations of time harmonic maxwell equations the key is to construct a special coarse mesh space which adds the kernel of the curl operator in a fine space to a coarse mesh space to solve the original problem and then uses the fine mesh space to solve the h curl elliptic problem it is shown that the generalized minimal residual gmres method applied to the preconditioned system converges uniformly provided that the coarsest mesh size is reasonably small but independent of the fine mesh size and the parameter for the coarse mesh space solver is sufficiently large numerical experiments show the efficiency of the proposed approach

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

a delta regularization finite element method for a double curl problem with divergence free constraint to deal with the divergence free constraint in a double curl problem rm curl mu 1 rm curl u f and rm div varepsilon u 0 in omega where mu and varepsilon represent the physical properties of the materials occupying omega we develop a delta regularization method rm curl mu 1 rm curl u delta delta varepsilon u delta f to completely ignore the divergence free constraint rm div varepsilon u 0 we show that u delta converges to u in h rm curl omega norm as delta rightarrow 0 the edge finite element method is then analyzed for solving u delta with the finite element solution u delta h a quasioptimal error bound in the h rm curl omega norm is obtained between u and u delta h including a uniform with respect to delta stability of u delta h in the h rm curl omega norm all the theoretical analysis is done in a general setting where mu and varepsilon may be discontinuous an

full wave analysis of dielectric waveguides at a given frequency new variational formulation to compute propagation constants is proposed based on it vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property convergence analysis is conducted in the framework of collectively compact operators reported theoretical results apply to a wide class of vector finite elements including two families of nedelec and their generalization the hp edge elements numerical experiments fully support theoretical estimates for convergence rates

global well posedness of the critical burgers equation in critical besov spaces abstract we make use of the method of modulus of continuity a kiselev f nazarov r shterenberg blow up and regularity for fractal burgers equation dyn partial differ equ 5 2008 211 240 and fourier localization technique h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 to prove the global well posedness of the critical burgers equation t u u x u u 0 in critical besov spaces b p 1 1 p r with p 1 where

a discrete weighted helmholtz decomposition and its application we propose a discrete weighted helmholtz decomposition for edge element functions the decomposition is orthogonal in a weighted l 2 inner product and stable uniformly with respect to the jumps in the discontinuous weight function as an application the new helmholtz decomposition is applied to demonstrate the quasi optimality of a preconditioned edge element system for solving a saddle point maxwell system in non homogeneous media by a non overlapping domain decomposition preconditioner i e the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain and more importantly it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media numerical experiments are presented to validate the effectiveness of the non overlapping domain decomposition preconditioner

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

the mortar finite element method for 3d maxwell equations first results in the framework of domain decomposition methods we extend the main ideas of the mortar element method to the numerical solution of maxwell s equations in wave form by h curl conforming finite elements the method we propose turns out to be a new nonconforming nonoverlapping domain decomposition method where nonmatching grids are allowed at the interfaces between subdomains a model problem is considered the convergence of the discrete approximation is analyzed and an error estimate is provided the method is proven to be slightly suboptimal with a loss of a factor scriptstyle sqrt rm ln h with respect to the degree of polynomials in order to achieve this convergence result we nevertheless need extra regularity assumptions on the solution of the continuous problem

on electric and magnetic problems for vector fields in anisotropic nonhomogeneous media r 3 2 initiated by saranen 131 in the above n is the outward drawn unit normal to the boundary and a denotes the exterior product according to the simple models for static magnetic fields resp electric fields which are governed by 0 1 resp 0 2 we call 0 1 the magnetic type problem and 0 2 the electric type problem considering bounded smooth domains a c r3 we discussed in 131 by means of an appropriate hilbert space method the solvability and the representation of the solutions for both problems 0 1 and 0 2 such a new approach was necessary to cover the general nonhomogeneous cases where v and e are matrix valued functions here our aim is twofold first we will now include exterior domains in our consideration this will be achieved by using certain weighted spaces which slightly restrict the behavior of the fields at infinity to find the scalar potentials we have to solve the dirichlet and neumann problems in exterior domains consequently in order to define the vector potentials we solve appropriate auxiliary second order boundary problems for vector fields the other aspect of this paper is to point out that the dimension of the space of solutions for 0 1 or 0 2 with homogeneous right side is 254 0022 247x 83 010254 22 03 00 0

convergence and optimality of adaptive edge finite element methods for time harmonic maxwell equations we consider a standard adaptive edge finite element method aefem based on arbitrary order nedelec edge elements for three dimensional indefinite time harmonic maxwell equations we prove that the aefem gives a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops provided the initial mesh is fine enough using the geometric decay we show that the aefem yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom the main technical contribution of the paper is in the establishment of a quasi orthogonality and a localized a posteriori error estimator

numerical analysis of a penalty approach for the solution of a transient eddy current problem abstract the aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity space discretization is based on nedelec edge elements while a backward euler scheme is used for time discretization the curl free constraint in the dielectric domain is imposed by means of a penalty strategy convergence of the penalized problem as the penalty parameter goes to zero is proved for the continuous and the discrete problems for the latter uniformly in the discretization parameters optimal order error estimates for the convergence of the discrete penalized problem with respect to the penalty and the discretization parameters are also proved finally some numerical tests are reported to assess the performance of this approach

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

boundary edge elements and spanning tree technique in three dimensional electromagnetic field computation the boundary discretization of the curl free vector variable such as the magnetic field h and the divergencefree variable such as the surface current density k is discussed instead of introducing the scalar variables the vector variables h and k are discretized by the curl conform and the div conform triangular edge elements respectively the degrees of freedom are associated with the boundary edges in order to ensure the null curl of h and the null divergence of k a spanning tree technique is used to identify the independent edges the triangular edge elements contain the first order nodal elements when expressing h or k by the scalar variables

an optimal domain decomposition preconditioner for low frequency time harmonic maxwell equations the time harmonic maxwell equations are considered in the low frequency case a finite element domain decomposition approach is proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure which is proven to converge the rate of convergence turns out to be independent of the mesh size showing that the preconditioner implicitly defined by the iterative procedure is optimal for obtaining this convergence result it has been necessary to prove a regularity theorem for dirichlet and neumann harmonic fields

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf 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 due the non density 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

a semi implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations abstract we present and analyze a semi implicit finite element method fem for the dynamical incompressible magnetohydrodynamics mhd equations the finite element approximation is based on mixed conforming elements where taylor hood type elements are used for the navier stokes equations and nedelec edge elements are used for the magnetic equation the divergence free conditions are weakly satisfied at the discrete level due to the use of nedelec edge element the proposed method is particularly suitable for problems defined on non smooth and multi connected domains for the temporal discretization we use a semi implicit scheme which only needs to solve a linear system at each time step moreover the linearized mixed fem is energy conserving we establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries numerical results are provided to show its effectiveness and verify the theoretical analysis in particular a benchmark lid driven cavity problem is provided to show that the proposed numerical method produces results as good as that of the ones which are divergence free

limits of weight spaces lusztig s q analogs and fiberings of adjoint orbits let g be a connected complex semisimple algebraic group and t a maximal torus inside a borel subgroup b with g t and b their lie algebras let v be a representation in the category a for g the t decomposition v met va of v into a direct sum of finite dimensional weight spaces is central in the representation theory of g in this paper we introduce on weight spaces a new structure the principal filtration je v where e is a principal nilpotent in g chosen to be compatible with t for example e can be the sum of the simple root vectors relative to t b this filtration is constructed in a very simple way by taking j v to be the space of vectors annihilated by the p 1 th power of e for p 0 our approach is motivated by kostant s fundamental work ki k2 on actions of the principal tds three dimensional subalgebra and coordinate rings of regular adjoint orbits in theorem 3 4 we give a new description in terms of the dimension jumps of the principal filtration of the weight space of lusztig s l q analog mm q of dominant weight multiplicity in a finite dimensional irreducible g representation v the polynomial m q was defined algebraically as an alternating sum over the weyl group through a q analog of kostant s weight multiplicity formula we prove that m q is equal to the jump polynomial

a domain decomposition approach for heterogeneous time harmonic maxwell equations the time harmonic maxwell equations for a composite medium which behaves like a conductor in one part and like a perfect insulator in the other one are considered an existence and uniqueness theorem is proven for this degenerate problem in the case of dirichlet boundary conditions a finite element domain decomposition approach is then proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure where at each step a non degenerate boundary value problem has to be solved in each subdomain the convergence of these iterations is proven and the rate of convergence turns out to be independent of the mesh size h showing that the preconditioner implicitly defined by the iteration procedure is optimal

an iterative substructuring method for maxwell s equations in two dimensions iterative substructuring methods also known as schur complement methods form an important family of domain decomposition algorithms they are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner for conforming finite element approximations of h 1 it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure in this paper the same result is established for similar iterative methods for low order n e d e lec finite elements which approximate hcurl in two dimensions results of numerical experiments are also provided

homology and cohomology computation in finite element modeling a homology and cohomology solver for finite element meshes is represented it is an integrated part of the finite element mesh generator gmsh we demonstrate the exploitation of the cohomology computation results in a finite element solver and use an induction heating problem as a working example the homology and cohomology solver makes the use of a vector scalar potential formulation straightforward this gives better overall performance than a vector potential formulation cohomology computation also clarifies the lumped parameter coupling of the problem and enables the user to obtain useful postprocessing data as a part of the finite element solution

a feti domain decomposition method for edge element approximations in two dimensions with discontinuous coefficients a class of finite element tearing and interconnecting feti methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed first an abstract framework is presented for the analysis of a class of feti methods where a natural coarse problem associated with the substructures is lacking then a family of feti methods for edge element approximations is proposed it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures the estimate is also independent of the jumps of both of the coefficients of the original problem numerical results validating our theoretical bounds are given the method and its analysis can be easily generalized to raviart thomas element approximations in two and three dimensions