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

convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

superconvergence analysis of yee scheme for metamaterial maxwell s equations on non uniform rectangular meshes since the development of yee scheme back in 1966 it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations however its rigorous error analysis on nonuniform rectangular type grids was carried out until 1994 by monk and suli they showed that the yee scheme is still second order convergent on a nonuniform mesh even though the local truncation error is only of first order in this paper we extend their results to maxwell s equations in metamaterials by a simpler proof and show the second order superconvergence in space for the true yee scheme instead of the only semi discrete form discussed in monk and suli s original work numerical results supporting our analysis are presented

an adaptive finite element method for the h ψ formulation of time dependent eddy current problems in this paper we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the h formulation of eddy current problems with multiply connected conductors multiply connected domains are considered by making cuts the competitive performance of the method is demonstrated by an engineering benchmark problem team workshop problem 7 and a singular problem with analytic solution

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

mortar formulation for a class of staggered discontinuous galerkin methods abstract a mortar formulation is developed and analyzed for a class of staggered discontinuous galerkin sdg methods applied to second order elliptic problems in two dimensions the computational domain consists of nonoverlapping subdomains and a triangulation is provided for each subdomain which need not conform across subdomain interfaces this feature allows a more flexible design of discrete models for problems with complicated geometries shocks or singular points a mortar matching condition is enforced on the solutions across the subdomain interfaces by introducing a lagrange multiplier space moreover optimal convergence rates in both l 2 and discrete energy norms are proved numerical results are presented to show the performance of the method

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

discontinuous galerkin computation of the maxwell eigenvalues on simplicial meshes this paper is concerned with the discontinuous galerkin approximation of the maxwell eigenproblem after reviewing the theory developed in a buffa i perugia discontinuous galerkin approximation of the maxwell eigenproblem technical report 24 pv imati cnr pavia italy 2005 we present a set of numerical experiments which both validate the theory and provide further insight regarding the practical performance of discontinuous galerkin methods particularly in the case when non conforming meshes characterized by the presence of hanging nodes are employed

towards the canonical tensor operators of uq 3 i the maximal null space case generalizing the su 3 canonical tensor operator concept biedenharn and louck to the quantum algebra uq 3 the wigner clebsch gordan coefficients of uq 3 with repeating irreducible representations are considered extremal projectors of the quantum algebra uq 3 in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq 3 canonical isoscalar factors explicit expressions of the uq 3 isofactors corresponding to the maximal null space case of the uq 3 unit canonical tensor operators and their normalization factors denominator functions are presented the transposition and conjugation phase factors for the su 3 and uq 3 canonical isofactors are correlated with phases and zeros of boundary isofactors invariance of the canonical isofactors or absence of such invariance under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance or numerical degeneracy of the usual splitting

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

dispersion analysis of finite difference and discontinuous galerkin schemes for maxwell s equations in linear lorentz media abstract in this paper we consider maxwell s equations in linear dispersive media described by a single pole lorentz model for electronic polarization we study two classes of commonly used spatial discretizations finite difference methods fd with arbitrary even order accuracy in space and high spatial order discontinuous galerkin dg finite element methods both types of spatial discretizations are coupled with second order semi implicit leap frog and implicit trapezoidal temporal schemes by performing detailed dispersion analysis for the semi discrete and fully discrete schemes we obtain rigorous quantification of the dispersion error for lorentz dispersive dielectrics in particular comparisons of dispersion error can be made taking into account the model parameters and mesh sizes in the design of the two types of schemes this work is a continuation of our previous research on energy stable numerical schemes for nonlinear dispersive optical media 6 7 the results for the numerical dispersion analysis of the reduced linear model considered in the present paper can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models the numerical dispersion analysis of the fully discrete fd and dg schemes for the dispersive maxwell model considered in this paper clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations their order of accuracy mesh discretization parameters and model parameters the results obtained here cannot be arrived at by considering discretizations of maxwell s equations in free space in particular our results contrast the advantages and disadvantages of using high order fd or dg schemes and leap frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors finally we highlight the limitations of the second order accurate temporal discretizations considered

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

a recursive approach to local mesh refinement in two and three dimensions abstract the newest vertex strategy for local refinement in two dimensions is reviewed and discussed a recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described the recursive approach requires certain restrictions on the initial mesh on the other hand under this condition the refinement process can be kept as local as needed and it can be fully inverted simple data structures and derefinement algorithms are also outlined

convergence analysis of a finite volume method for maxwell s equations in nonhomogeneous media in this paper we analyze a recently developed finite volume method for the time dependent maxwell s equations in a three dimensional polyhedral domain composed of two dielectric materials with different parameter values for the electric permittivity and the magnetic permeability convergence and error estimates of the numerical scheme are established for general nonuniform tetrahedral triangulations of the physical domain in the case of nonuniform rectangular grids the scheme converges with second order accuracy in the discrete l2 norm despite the low regularity of the true solution over the entire domain in particular the finite volume method is shown to be superconvergent in the discrete h curl norm in addition the explicit dependence of the error estimates on the material parameters is given

stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics abstract we analyze the stability of a linearized hydrodynamical model describing the response of nanometric dispersive metallic materials illuminated by optical light waves that is the situation occurring in nanoplasmonics this model corresponds to the coupling between the maxwell system and a pde describing the evolution of the polarization current of the electrons in the metal we show the well posedness of the system polynomial stability and optimal energy decay rate we also investigate the numerical stability for a discontinuous galerkin type approximation and several explicit time integration schemes

convergence analysis of adaptive edge finite element method for variable coefficient time harmonic maxwell s equations abstract in this paper our main goal is to study the convergence analysis of adaptive edge finite element method aefem based on arbitrary order nedelec edge elements for the variable coefficient time harmonic maxwell s equations i e we prove that the aefem gives a contraction for the sum of the energy error and the error estimator between two consecutive adaptive loops provided the initial mesh is fine enough first we give the variational problem of the variable coefficient time harmonic maxwell s equations and the posteriori error estimator of the residual type then we establish the quasi orthogonality the global upper bound of the error the compressibility of the error estimator and prove the convergence result finally our numerical results verify that the error estimator is valid

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

symmetry properties of matrix elements of canonical su 3 tensor operators the symmetries of the su 3 3 j symbols which are defined as symmetrized matrix elements of the canonical su 3 tensor operators are investigated the symmetries considered are those which in su 2 correspond to the interchange of columns of the 3 j symbol as well as the symmetry under conjugation it is found that for each tensor operator in a multiplicity set the matrix elements for a fixed operator pattern carry a one dimensional representation of the symmetric group s3

interior penalty dg methods for maxwell s equations in dispersive media in this paper we develop a fully discrete interior penalty discontinuous galerkin method for solving the time dependent maxwell s equations in dispersive media the model is described by a vector integral differential equation our scheme is proved to be unconditionally stable and achieve optimal error estimates in both l 2 norm and energy norm the scheme is implemented and numerical results supporting our analysis are presented

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

determination of a time dependent convolution kernel from a boundary measurement in nonlinear maxwell s equations a nonlinear hyperbolic maxwell equation with an unknown solely time dependent convolution kernel is studied the missing kernel is recovered from an additional normal component measurement over the whole boundary the existence of a solution to this inverse problem is shown moreover a constructive algorithm for approximations is designed and its convergence is established uniqueness is proved for a regular solution theoretical results are supported by a numerical experiment

well posedness study for a time domain spherical cloaking model in this paper we study a time domain spherical cloaking model recently introduced by zhao and hao 2009 this model is quite complicated and is composed of four coupled differential equations here we first prove the existence and uniqueness of a solution for this model then we obtain a stability result which analysis is quite involved due to the coupling between the four variables to our best knowledge this is the first well posedness study carried out for the time domain cloaking model with metamaterials

a deluxe feti dp algorithm for a hybrid staggered discontinuous galerkin method for h curl elliptic problems summary convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered dg finite element approximation of h curl elliptic problems in two dimensions in addition to the advantages of staggered dg methods the basis functions of the new hybrid staggered dg method are all locally supported in the triangular elements and a lagrange multiplier approach is applied to enforce the global connections of these basis functions the interface problem on the lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem our algorithm is shown to give a condition number bound ofc 1c log h h 2 independent of the two parameters whereh his the number of triangles across each subdomain numerical results are included to confirm our theoretical bounds copyright 2013 john wiley sons ltd received 12 september 2013 accepted 11 november 2013 in this paper a dual and primal finite element tearing and interconnecting feti dp algorithm is developed for a fast and stable solution of a new hybrid staggered dg hsdg method applied to h curl problems staggered dg sdg methods were first introduced in 1 3 for wave propagation problems the idea was subsequently applied to other problems such as convection diffusion equations 4 electromagnetic problems 5 9 stokes equations 10 and multiscale wave simulations 11 12 similar to 5 the advantages of using sdg for h curl problems are the preservation of the structures of the differential operators the local conservation property and the optimal convergence in particular the discretizations of the two curl operators by the sdg method are adjoint to each other and the null space of the discrete curl operator is exactly the gradients moreover the divergence condition is automatically satisfied in an appropriate weak sense despite the aforementioned advantages the implementation of sdg methods requires careful numbering of the supports of the basis functions to allow an easier implementation of the sdg method the hsdg method is introduced in this paper one distinctive advantage of the hsdg method is that the basis functions are all locally supported in the triangular elements hence the numbering can be easily performed the fact that the basis functions are totally discontinuous requires some global couplings in particular we will use the lagrange multiplier approach to enforce the global connection of the basis functions this idea is also related to the hybridized dg methods 13 16 in fact it is shown that the sdg method is the limit of a hybridized dg method

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

mathematical and numerical study of wave propagation in negative index materials abstract in this paper we develop a crank nicolson mixed finite element method for modeling wave propagation in negative index materials nims the nims model is formed as a time dependent system involving four dependent vector variables the electric and magnetic fields and the induced electric and magnetic currents optimal error estimates for all four variables are proved for nedelec edge elements numerical examples are presented to show the exotic properties for wave propagation in nims

superconvergent recovery of edge finite element approximation for maxwell s equations abstract a new recovery method of edge elements for maxwell s equations is proposed to reduce the recovery problem into one dimensional settings by using the local symmetry projection the recovered method is applied to both nedelec interpolation and edge finite element approximation superconvergence results are established for both the postprocessed nedelec interpolation and the recovered edge finite element approximation in a discrete norm numerical examples are presented to illustrate our theoretical analysis

on sequential locally repairable codes we consider the locally repairable codes lrcs aiming at sequentially recovering multiple erasures in particular we propose and study the so called n k r t sequential lrcs slrc as an n k linear code where any t leq t erasures can be sequentially recovered each by r 2 leq r other code symbols here sequential recovering means that the erased symbols are recovered one by one and an already recovered symbol can be used to recover the remaining erased symbols this important recovering method in contrast with the extensively studied parallel recovering is currently far from being thoroughly understood more specifically there are to date no codes constructed for arbitrary t geq 3 erasures and bounds to evaluate the performance of such codes we first derive a tight upper bound on the code rate of the n k r t slrc for t 3 and r geq 2 we then propose two constructions of binary n k r t slrcs for general r t geq 2 existing constructions only deal with t leq 7 erasures the first construction generalizes the method of direct product construction the second construction is based on the resolvable configurations and yields slrcs for any r geq 2 and odd t geq 3 for both constructions the rates are optimal for t in 2 3 and are higher than most of the existing lrc families for arbitrary t geq 4

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

convergence analysis of fully discrete finite volume methods for maxwell s equations in nonhomogeneous media we will consider both explicit and implicit fully discrete finite volume schemes for solving three dimensional maxwell s equations with discontinuous physical coefficients on general polyhedral domains stability and convergence for both schemes are analyzed we prove that the schemes are second order accurate in time both schemes are proved to be first order accurate in space for the voronoi delaunay grids and second order accurate for nonuniform rectangular grids we also derive explicit expressions for the dependence on the physical parameters in all estimates

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

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

analysis and application of the nodal discontinuous galerkin method for wave propagation in metamaterials in this paper we develop a nodal discontinuous galerkin method for solving the time dependent maxwell s equations when metamaterials are involved both semi and fully discrete schemes are constructed numerical stability and error estimate are proved for both schemes numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial maxwell s equations

an a priori error estimate for a temporally discontinuous galerkin space time finite element method for linear elasto and visco dynamics the work of shaw was supported in part by the engineering and physical sciences research council ep h011072 1 this support is gratefully acknowledged

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

an adaptive edge finite element method for electromagnetic cloaking simulation in this paper we develop an adaptive edge finite element method based on a reliable and efficient recovery type a posteriori error estimator for time harmonic maxwell equations the asymptotically exact a posteriori error estimator is based on the superconvergence result proved for the lowest order edge element on triangular grids where most pairs of triangles sharing a common edge form approximate parallelograms the efficiency and robustness of the proposed method is demonstrated by extensive numerical experiments for electromagnetic cloaking problems with highly anisotropic permittivity and permeability

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

a short note on plain convergence of adaptive least squares finite element methods abstract we show that adaptive least squares finite element methods driven by the canonical least squares functional converge under weak conditions on pde operator mesh refinement and marking strategy contrary to prior works our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters finally we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions e g the preconditioned conjugate gradient method with optimal preconditioner the results apply within a fairly abstract framework which covers a variety of model problems

a staggered discontinuous galerkin method for wave propagation in media with dielectrics and meta materials some electromagnetic materials exhibit in a given frequency range effective dielectric permittivity and or magnetic permeability which are negative in the literature they are called negative index materials left handed materials or meta materials we propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta material the method we investigate can be considered as an alternative method compared to the method presented by the second author and co workers in particular we shall use the abstract framework they developed to prove well posedness of the exact problem we recast this problem to fit later discretization by the staggered discontinuous galerkin method developed by the first author and co worker a method which relies on introducing an auxiliary unknown convergence of the numerical method is proven with the help of explicit inf sup operators and numerical examples are provided to show the efficiency of the method

overlapping schwarz preconditioners for indefinite time harmonic maxwell equations time harmonic maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite a galerkin method using nedelec spaces can be employed to get approximate solutions numerically the problem of preconditioning the indefinite matrix arising from this method is discussed here specifically two overlapping schwarz methods will be shown to yield uniform preconditioners

locally repairable codes joint sequential parallel repair for multiple node failures locally repairable codes lrc have been studied from two approaches to locally repair multiple failed nodes 1 parallel approach in which a coordinate i of an n k d linear code is said to have locality r and availability t if there exist t disjoint repair sets each of which contains at most r other coordinates that can recover the value of the i th coordinate 2 sequential approach in which the erased symbols failed nodes are repaired one by one and any previously repaired node can be used to repair the remaining failed nodes in this paper we first consider lrc aiming at joint sequential parallel repairing multiple failed nodes and study the n k r t u elrcs exact locally repairable codes which are n k linear codes with the property that any set of failed nodes of size at most t can be simultaneously repaired in parallel mode and each element of a set e of failed nodes of size at most u can be sequentially repaired by r r other coordinates we present a method by which with a given parity check matrix of an n k r t u elrc with minimum hamming distance d a new elrc with minimum hamming distance 2d and availability t 1 is constructed that can repair each set of failed nodes e of size at most 2u 1 in sequential mode and this repair is done in at most u t 2 steps we construct a big family of lrcs by making use of orthogonal latin rectangles and permutation cubes and some other combinatorial designs the constructed codes contain the family of direct product codes we also use m dimensional permutation cubes to construct lrcs with short block length for each r

two and three field formulations for wave transmission between media with opposite sign dielectric constants we consider a simplified scalar model problem related to maxwell equations involving wave transmission between media with opposite sign dielectric and or magnetic constants we build two variational formulations equivalent to the model problem we show that under some suitable conditions both formulations are well posed since they fit into the coercive plus compact framework advantages over previous studies is the validity of the formulations in the general case of lipschitz interface between the two media and l dielectric and magnetic constants an interesting feature of these formulations is that they allow a simple finite element numerical implementation

large scale 3d geoelectromagnetic modeling using parallel adaptive high order finite element method abstractwe have investigated the use of the adaptive high order finite element method fem for geoelectromagnetic modeling because high order fem is challenging from the numerical and computational points of view most published finite element studies in geoelectromagnetics use the lowest order formulation solution of the resulting large system of linear equations poses the main practical challenge we have developed a fully parallel and distributed robust and scalable linear solver based on the optimal block diagonal and auxiliary space preconditioners the solver was found to be efficient for high finite element orders unstructured and nonconforming locally refined meshes a wide range of frequencies large conductivity contrasts and number of degrees of freedom dofs furthermore the presented linear solver is in essence algebraic i e it acts on the matrix vector level and thus requires no information about the discretization boundary conditions or physical source used making it readily ef

developing a time domain finite element method for modeling of electromagnetic cylindrical cloaks in this paper we propose a time domain finite element method for modeling of electromagnetic cloaks the permittivity and permeability of the cloak model are described by the drude dispersion model the model to be solved is quite challenging in that we have to solve a coupled problem with different partial differential equations given in different regions our method is based on a mixed finite element method using edge elements with different types of meshes in different regions numerical results demonstrate that our algorithm is quite effective for simulating cloaks in time domain to our knowledge this is the first cloak simulation carried out by the time domain finite element method

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

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

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

solving metamaterial maxwell s equations via a vector wave integro differential equation in this paper we discuss the time domain metamaterial maxwell s equations one major contribution of this paper is that after some effort we find that the metamaterial maxwell s equations can be beautifully reduced to a vector wave integro differential equation involving just one unknown which is quite similar to that obtained from the standard maxwell s equations in vacuum then we study the existence and uniqueness of this new modeling equations and propose a fully discrete finite element method to solve this model numerical results justifying our analysis are presented this discovery shall make simulation of metamaterials much more efficient than the previous works

multiscale computations for 3d time dependent maxwell s equations in composite materials we discuss multiscale computations of the initial boundary value problem for three dimensional time dependent maxwell s equations in composite materials a multiscale algorithm is developed and the convergence results are presented numerical simulations are carried out to validate the theoretical results reported in the present paper

symmetry properties of s classified su 3 3j 6j and 9j symbols starting from the s classified su 3 clebsch gordan coefficients introduced previously pluhar et al 1985 the s classified su 3 3j 6j and 9j symbols are constructed and examined they satisfy simple symmetry relations similar to those valid for the su 2 case

convergence and quasi optimality of adaptive finite element methods for harmonic forms numerical computation of harmonic forms typically called harmonic fields in three space dimensions arises in various areas including computer graphics and computational electromagnetics the finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms in this work we study the convergence properties of adaptive finite element methods afem for computing harmonic forms we show that a properly defined afem is contractive and achieves optimal convergence rate beginning from any initial conforming mesh this result is contrasted with related afem convergence results for elliptic eigenvalue problems where the initial mesh must be sufficiently fine in order for afem to achieve any provable convergence rate

discontinuous galerkin methods for maxwell s equations in drude metamaterials on unstructured meshes abstract in this follow up work we extend the discontinuous galerkin dg methods previously developed on rectangular meshes li et al 2017 to triangular meshes the dg schemes in li et al 2017 are both optimally convergent and energy conserving however as we shall see in the numerical results section the dg schemes on triangular meshes only have suboptimal convergence rate we prove the energy conservation and an error estimate for the semi discrete schemes the stability of the fully discrete scheme is proved and its error estimate is stated we present extensive numerical results with convergence consistent of our error estimate and simulations of wave propagation in drude metamaterials to demonstrate the flexibility of triangular meshes

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

theoretical and numerical analysis of a non local dispersion model for light interaction with metallic nanostructures in this paper we discuss the time domain maxwell s equations coupled to another partial differential equation which arises from modeling of light and structure interaction at the nanoscale one major contribution of this paper is that the well posedness is rigorously justified for the first time then we propose a fully discrete finite element method to solve this model it is interesting to note that we need use curl conforming divergence conforming and l 2 finite elements for this model numerical stability and optimal error estimate of the scheme are proved numerical results justifying our theoretical convergence rate are presented

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

a second order numerical scheme for nonlinear maxwell s equations using conforming finite element abstract in this paper we study and analyze a second order numerical scheme for maxwell s equations with nonlinear conductivity using the nedelec finite element method fem a purely explicit treatment of the nonlinear term greatly simplifies the computational effort since we only need to solve a constant coefficient linear system at each time step the curl conforming nature of the nedelec element assures its divergence free property in turn we present the linearized stability analysis for the numerical error function to obtain an optimal l2 error estimate in more details an o 2 h s error estimate in the l2 norm yields the maximum norm bound of the numerical solution so that the convergence analysis could be carried out at the next time step a few numerical examples in the transverse electric te case in two dimensional spaces are also presented which demonstrate the efficiency and accuracy of the proposed numerical scheme

reconstruction of a time dependent source term from a single boundary measurement in maxwell s equations with nonlinear generalized ohm s law abstract hyperbolic maxwell s equation with an unknown time dependent source is investigated we consider a nonlinear generalized ohm s law in our model the source is reconstructed from a single boundary measurement over a part of the boundary we use well known rothe s method to show the existence of a solution in the case of regular solution we provide a uniqueness proof as well to support theoretical results a numerical experiment is provided

smoothed projections in finite element exterior calculus the development of smoothed projections constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator have proved to be an eective tool in finite element exterior calculus the advan tage of these operators is that they are l 2 bounded projections and still they commute with the exterior derivative in the present paper we generalize the construction of these smoothed projections such that also non quasi uniform meshes and essential boundary conditions are covered the new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

a priori and posteriori error analysis for time dependent maxwell s equations abstract we consider time dependent maxwell s equations discretized by variable time steps in time domain and edge elements in spatial domain first the stability and optimal a priori error estimate are proved for both semi and fully discrete schemes then a posteriori error analysis is carried out for both schemes

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

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

a posteriori error analysis for a staggered discontinuous galerkin discretization of the time harmonic maxwell s equations in this paper we present the first a posteriori error analysis for the staggered discontinuous galerkin sdg method specifically we consider the approximation of the time harmonic maxwell s equations by a sdg method and prove that our residual based a posteriori error indicator is both reliable and efficient we validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems

convergence of an aefem for time harmonic maxwell equations with variable coefficients abstract an adaptive edge finite element method aefem for three dimensional indefinite time harmonic maxwell equations with variable coefficients is proved to be convergent and arbitrary order nedelec edge elements are considered a posteriori upper bound quasi orthogonality and the contraction of the error estimator are provided to prove the contraction of the sum of the energy error and the scaled error estimator between two consecutive adaptive loops numerical experiments are presented to support the theoretical results

an adaptive least squares fem for the stokes equations with optimal convergence rates this paper introduces the first adaptive least squares finite element method ls fem for the stokes equations with optimal convergence rates based on the newest vertex bisection with lowest order raviart thomas and conforming p 1 p1 discrete spaces for the divergence least squares formulation in 2d although the least squares functional is a reliable and efficient error estimator the novel refinement indicator stems from an alternative explicit residual based a posteriori error control with exact solve particular interest is on the treatment of the data approximation error which requires a separate marking strategy the paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non linear approximation class it extends and generalizes the approach of carstensen and park siam j numer anal 53 43 62 2015 from the poisson model problem to the stokes equations

a posteriori error indicators for maxwell s equations abstract in this paper we shall analyze a class of a posteriori error indicators for an electromagnetic scattering problem for maxwell s equations in the presence of a bounded inhomogeneous and anisotropic scatterer problems of this type arise when computing the interaction of electromagnetic radiation with biological tissue we briefly recall existence and uniqueness theory associated with this problem then we show how a posteriori error indicators can be derived using an adjoint equation approach the error indicators use both the jump in normal and tangential components of the field across faces in the mesh

discontinuous galerkin methods for dispersive and lossy maxwell s equations and pml boundary conditions in this paper we will present a unified formulation of discontinuous galerkin method dgm for maxwell s equations in linear dispersive and lossy materials of debye type and in the artificial perfectly matched layer pml regions an auxiliary differential equation ade method is used to handle the frequency dependent constitutive relations with the help of auxiliary polarization currents in the computational and pml regions the numerical flux for the dispersive lossy maxwell s equations with the auxiliary polarization current variables is derived various numerical results are provided to validate the proposed formulation

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

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

superconvergence of mixed finite element approximations to 3 d maxwell s equations in metamaterials numerical simulation of metamaterials has attracted more and more attention since 2000 after the first metamaterial with negative refraction index was successfully constructed in this paper we construct a fully discrete leap frog type finite element scheme to solve the three dimensional time dependent maxwell s equations when metamaterials are involved first we obtain some superclose results between the interpolations of the analytical solutions and finite element solutions obtained using arbitrary orders of raviart thomas nedelec mixed spaces on regular cubic meshes then we prove the superconvergence result in the discrete l 2 norm achieved for the lowest order raviart thomas nedelec space to our best knowledge such superconvergence results have never been obtained elsewhere finally we implement the leap frog scheme and present numerical results justifying our theoretical analysis

the staggered dg method is the limit of a hybridizable dg method we show in the framework of steady state diffusion boundary value problems that the staggered discontinuous galerkin sdg method siam j numer anal 47 2009 pp 3820 3848 can be obtained from a hybridizable discontinuous galerkin hdg method siam j numer anal 47 2009 pp 1319 1365 by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others we then show that this point of view allows the sdg method to immediately acquire new properties all inherited from the hdg methods namely their efficient implementation by hybridization their postprocessings and their superconvergence properties

numerical convergence and physical fidelity analysis for maxwell s equations in metamaterials abstract in this paper we develop a leap frog mixed finite element method for solving maxwell s equations resulting from metamaterials our scheme is similar to the popular yee s fdtd scheme used in electrical engineering community and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed our scheme is proved to obey the gauss s law automatically if the initial fields satisfy that furthermore the conditional stability and optimal error estimate for the proposed scheme are proved to our best knowledge we are unaware of any other publications devoted to the convergence analysis of this leap frog explicit scheme for maxwell s equations even in a simple medium while our results for metamaterials automatically reduce to the standard maxwell s equations in vacuum by dropping some terms resulting from the constitutive equations numerical results confirming our analysis are presented