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

going from cohomology to hochschild cohomology abstract the relation between ordinary cohomology and hochschild cohomology is investigated for quotients of quiver algebras which are either graded or finite dimensional it is shown that a resolution of the direct sum of the simple modules of the algebra can often be made two sided and yield a resolution of the algebra itself over its enveloping algebra the multiplicative structures of the cohomologies are shown to be related by a spectral sequence of algebras converging to the hochschild cohomology whose first term is given by a tensor product of the algebra itself with its cohomology algebra

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

fully automatic hp adaptivity for maxwell s equations i report on the development of a fully automatic hp adaptive strategy for the solution of time harmonic maxwell equations the strategy produces a sequence of grids that deliver exponential convergence for both regular and singular solutions given a coarse mesh we refine it first globally in both h and p and solve the problem on the resulting fine mesh we consider then the projection based interpolants of the fine mesh solution with respect to both current and next to be determined coarse grid and introduce the interpolation error decrease rate equal to the difference of the old and new coarse mesh interpolation errors vs number of degrees of freedom added the optimal hp refinements leading to the next coarse grid are then determined by maximizing the interpolation error decrease rate

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

bi relative algebraic k theory and topological cyclic homology it is well known that algebraic k theory preserves products of rings however in general algebraic k theory does not preserve fiber products of rings and one defines bi relative algebraic k theory so as to measure the deviation it was proved recently by cortinas 6 that rationally bi relative algebraic k theory and bi relative negative cyclic homology agree in this paper we show that with finite coefficients bi relative algebraic k theory and bi relative topological cyclic homology agree as an application we show that for a possibly singular curve over a perfect field k of positive characteristic p the cyclotomic trace map induces an isomorphism of the p adic algebraic k groups and the p adic topological cyclic homology groups in non negative degrees as a further application we show that the difference between the p adic k groups of the integral group ring of a finite group and the p adic k groups of a maximal z order in the rational group algebra can be expressed entirely in terms of topological cyclic homology let f be a functor that to a unital associative ring a associates a symmetric spectrum f a if i a is a two sided ideal the relative term f a i is defined to be the homotopy fiber of the map f a f a i induced from the canonical projection if further f a b is a ring homomorphism such that f i f i is an isomorphism onto a two sided ideal of b there is an induced map f a i f b f i and the bi relative term f a b i is defined to be the homotopy fiber of this map then there is a distinguished triangle in the stable homotopy category

dissipative terms and local time stepping improvements in a spatial high order discontinuous galerkin scheme for the time domain maxwell s equations in this paper we present some improvements in terms of accuracy and speed up for a particular well adapted discontinuous galerkin method devoted to the time domain maxwell equations first to reduce spurious modes on very distorted meshes the addition of dissipative terms as penalization in the numerical scheme is studied and compared on examples second in order to increase the efficiency of the method a multi class local time stepping strategy is presented and its validation and advantages are highlighted on different examples

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

finite element study of the lorentz model in metamaterials metamaterials are materials with electromagnetic properties that originate from artificial sub wavelength structures although the lorentz metamaterial model has been used by physicists and electrical engineers its mathematical properties have not been investigated yet also more robust and efficient algorithms for time domain simulations are needed in this paper we initiate our work in this respect by developing some finite element schemes for solving the lorentz model stability and error analysis are proved for these schemes and 2 d numerical results justifying our theoretical results are presented

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

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

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 well posed problem in singular fickian diffusion we prove that the problem of solving u t u m 1 u x x text for 1 m leqq 0 with initial conditionu x 0 x and flux conditions at infinity mathop lim limits x to infty u m 1 u x f t mathop lim limits x to infty u m 1 u x g t admits a unique solution u in c infty infty x infty 0 t t for every el1 r 0 0 and every pair of nonnegative flux functionsf g e l loc 0 the maximal existence time is given by t sup left t smallint phi x dx int limits 0 t f s g s ds right this mixed problem is ill posed for anym outside the above specified range

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

error analysis of finite element methods for 3 d maxwell s equations in dispersive media we consider the time dependent maxwell s equations in dispersive media on a bounded three dimensional domain a semi discrete standard finite element method and a semi discrete mixed finite element method are developed and error estimates are provided we believe this is the first finite element error analysis carried out for maxwell s equations in dispersive media

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

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

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

development and analysis of both finite element and fourth order in space finite difference methods for an equivalent berenger s pml model abstract this paper deals with an equivalent berenger s perfectly matched layer pml model we first develop a finite element scheme using edge elements to solve this model we prove a discrete stability of this method which inherits the stability obtained in the continuous case then we propose a fourth order in space finite difference scheme for solving this pml model numerical stability similar to the continuous stability and the optimal error estimate are established for the difference scheme here only second order time discretizations are considered for both schemes finally numerical results are presented to justify our analysis and demonstrate the effectiveness of this pml model for absorbing impinging waves

energy norm a posteriori error estimation for mixed discontinuous galerkin approximations of the maxwell operator in this paper we develop the a posteriori error estimation of mixed discontinuous galerkin finite element approximations of the maxwell operator in particular by employing suitable helmholtz decompositions of the error together with the conservation properties of the underlying method computable upper bounds on the error measured in terms of a natural mesh dependent energy norm are derived numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented

superconvergence analysis for maxwell s equations in dispersive media in this paper we consider the time dependent maxwell s equations in dispersive media on a bounded three dimensional domain global superconvergence is obtained for semi discrete mixed finite element methods for three most popular dispersive media models the isotropic cold plasma the one pole debye medium and the two pole lorentz medium global superconvergence for a standard finite element method is also presented to our best knowledge this is the first superconvergence analysis obtained for maxwell s equations when dispersive media are involved

local time space mesh refinement for simulation of elastic wave propagation in multi scale media this paper presents an original approach to local time space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro heterogeneities the main features of the algorithm are the application of temporal and spatial refinement on two different surfaces the use of the embedded stencil technique for the refinement of grid step with respect to time the use of the fast fourier transform fft based interpolation to couple variables for spatial mesh refinement the latter makes it possible to perform filtration of high spatial frequencies which provides stability in the proposed finite difference schemes in the present work the technique is implemented for the finite difference simulation of seismic wave propagation and the interaction of such waves with fluid filled fractures and cavities of carbonate reservoirs however this approach is easy to adapt and or combine with other numerical techniques such as finite elements discontinuous galerkin method or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations

stabilization and asymptotic behavior of dispersive medium models abstract we analyze the stability of different vectorial and scalar dispersive medium models set in a bounded domain of r d d 1 they correspond to the coupling between the maxwell system or the wave equation and some first order differential equations of parabolic type we show that some models have an exponential decay rate while the other ones are only polynomially stable in that last case we show that the obtained polynomial decay is optimal

a posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients we perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients according to bonnet bendhia et al 2010 9 if the contrast is large enough the continuous problem can be transformed into a coercive one we further show that a similar property holds for the discrete problem for any regular meshes extending the framework from bonnet bendhia et al 9 the reliability and efficiency of the proposed estimator are confirmed by some numerical tests

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

energy stable discontinuous galerkin methods for maxwell s equations in nonlinear optical media abstract the propagation of electromagnetic waves in general media is modeled by the time dependent maxwell s partial differential equations pdes coupled with constitutive laws that describe the response of the media in this work we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations odes which include a single resonance linear lorentz dispersion and the nonlinearity comes from the instantaneous electronic kerr response and the residual raman molecular vibrational response to design efficient accurate and stable computational methods we apply high order discontinuous galerkin discretizations in space to the hybrid pde ode maxwell system with several choices of numerical fluxes and the resulting semi discrete methods are shown to be energy stable under some restrictions on the strength of the nonlinearity error estimates are also established when we turn to fully discrete methods the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms to overcome this novel strategies are proposed to treat the nonlinearity in our model within the framework of the second order leap frog and implicit trapezoidal time integrators the performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves and third harmonic generation in soliton propagation

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

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

maxwellian material based absorbing boundary conditions absorbing boundary conditions abcs for the finite difference time domain fdtd method are introduced which arise from surrounding the simulation space with lossy maxwellian material layers generalizations of the standard lorentz dispersion material model the time derivative and two time derivative lorentz material models are developed for this purpose the advantages of this approach include the close connection of the abcs with the actual absorber physics associated with maxwell s equations the avoidance of the field equation splitting required by the berenger pml layers and reduced memory and operation counts several multi dimensional cases are presented to quantify the efficacy of this maxwellian material based approach

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

mathematical analysis and finite element simulation of a magnetized ferrite model microwave scattering by obstacles made of ferrite materials has been studied for a long time because of its important applications in radar satellite optical imaging system signal processing atmospheric science and even cloaking device design in this paper we present for the first time the rigorous mathematical analysis of a magnetized ferrite model widely used by engineers the well posedness of the model is established first then we develop a finite element time domain method for solving this model numerical stability similar to the continuous stability is proved for our scheme and the optimal error estimate in the l 2 norm is obtained numerical results are consistent with our theoretical analysis and show how wave propagates in the ferrite material

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

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

discontinuous viscosity solutions of a degenerate parabolic equation we study a nonlinear degenerate parabolic equation of the second order regularizing the equation by adding some artificial viscosity we construct a generalized solution we show that this solution is not necessarily continuous at all points

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

analysis of a time domain finite element method for 3 d maxwell s equations in dispersive media abstract we consider the time dependent maxwell s equations in dispersive media on a bounded domain in three dimensional space a fully discrete finite element scheme is developed to approximate the electric field equation derived from the maxwell s equations optimal energy norm error estimates are proved for nedelec curl conforming edge elements this is the first finite element error analysis for maxwell s equations in dispersive media

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

radial equivalence and study of self similarity for two very fast diffusion equations abstract in this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p laplacian equation begun in a previous work we treat the cases m 0 and p 1 we perform an exhaustive study of self similar solutions for both equations based on a phase plane analysis and the correspondences we discover we also obtain special correspondence relations and self maps for the limit case m 1 p 0 which is particularly important in applications in image processing we also find self similar solutions for the very fast p laplacian equation that have finite mass and in particular some of them that conserve mass while this phenomenon is not true for the very fast diffusion equation

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

optimal non dissipative discontinuous galerkin methods for maxwells equations in drude metamaterials simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain maxwells equations than the standard maxwells equations in free space in this paper we develop and analyze a non dissipative discontinuous galerkin dg method for solving the maxwells equations in drude metamaterials previous discontinuous galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non dissipative but sub optimal or dissipative and optimal our method uses a different and simple choice of numerical fluxes achieving provable non dissipative stability and optimal error estimates simultaneously we prove the stability and optimal error estimates for both semi and fully discrete dg schemes with the leap frog time discretization for the fully discrete case numerical results are given to demonstrate that the dg method can solve metamaterial maxwells equations effectively

time harmonic wave diffraction problems in materials with sign shifting coefficients some electromagnetic materials present in a given frequency range an effective dielectric permittivity and or magnetic permeability which are negative we are interested in the reunion of such a negative material and a classical one more precisely we consider here a scalar model problem for the simulation of a wave transmission between two such materials this model is governed by a helmholtz equation with a weight function in the d principal part which takes positive and negative real values introducing additional unknowns we have already proposed in bonnet ben dhia et al 2006 1 some new variational formulations of this problem which are of fredholm type provided the absolute value of the contrast of permittivities is large enough and therefore suitable for a finite element discretization we prove here that under similar conditions on the contrast the natural variational formulation of the problem although not coercive plus compact is nonetheless suitable for a finite element discretization this leads to a numerical approach which is straightforward less costly than the previous ones and very accurate

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

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

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

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

error analysis of mixed finite element methods for wave propagation in double negative metamaterials in this paper we develop both semi discrete and fully discrete mixed finite element methods for modeling wave propagation in three dimensional double negative metamaterials optimal error estimates are proved for nedelec spaces under the assumption of smooth solutions to our best knowledge this is the first error analysis obtained for maxwell s equations when metamaterials are involved

an implicit leap frog discontinuous galerkin method for the time domain maxwell s equations in metamaterials numerical simulation of metamaterials play a very important role in the design of invisibility cloak and sub wavelength imaging in this paper we propose a leap frog discontinuous galerkin method to solve the time dependent maxwell s equations in metamaterials conditional stability and error estimates are proved for the scheme the proposed algorithm is implemented and numerical results supporting the analysis are provided

energy conserved splitting spectral methods for two dimensional maxwell s equations in this paper the energy conserved splitting legendre galerkin method and energy conserved splitting legendre collocation method for maxwell s equations in two dimensions are proposed the schemes are energy conserved unconditionally stable and can be implemented efficiently the both methods are of second order convergence in time the high order error estimates in space for the two methods are also obtained for the ec s lc method an identity of the discrete divergence of electric fields is also obtained numerical examples are provided to confirm the theoretical analysis

infinitesimal cohomology and the chern character to negative cyclic homology there is a chern character from k theory to negative cyclic homology we show that it preserves the decomposition coming from adams operations at least in characteristic zero

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

a convergence analysis of yee s scheme on nonuniform grids the yee scheme is the principal finite difference method used in computing time domain solutions of maxwell s equations on a uniform grid the method is easily seen to be second order convergent in space this paper shows that the yee scheme is also second order convergent on a nonuniform mesh despite the fact that the local truncation error is nodally only of first order

development of discontinuous galerkin methods for maxwell s equations in metamaterials and perfectly matched layers the discontinuous galerkin method has proved to be an accurate and efficient way to numerically solve many differential equations in this paper we extend this method to solve the time dependent maxwell s equations when metamaterials and perfectly matched layers are involved numerical results are presented to demonstrate that our method is not only simple to implement but also quite effective in solving maxwell s equations in complex media

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

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

interior penalty discontinuous galerkin method for maxwell s equations energy norm error estimates we develop the symmetric interior penalty discontinuous galerkin dg method for the time dependent maxwell equations in second order form we derive optimal a priori error estimates in the energy norm for smooth solutions we also consider the case of low regularity solutions that have singularities in space

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

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

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

mathematical analysis of ziolkowski s pml model with application for wave propagation in metamaterials abstract in this paper we investigate one perfectly matched layer pml model proposed by ziolkowski in 1999 various schemes for solving this pml model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem however a rigorous analysis of this model is lacking in this paper we establish the stability of this pml model and propose a fully discrete finite element scheme to solve this model with edge elements discrete stability and optimal error estimate of this scheme are proved numerical results justifying the analysis and demonstrating the effectiveness of this pml model are presented

local smoothing effects positivity and harnack inequalities for the fast p laplacian equation we study qualitative and quantitative properties of local weak solutions of the fast p laplacian equation tu pu with 1 p 2 our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of rn 0 t we combine these lower and upper bounds in different forms of intrinsic harnack inequalities which are new in the very fast diffusion range that is when 1 p 2n n 1 the boundedness results may be also extended to the limit case p 1 while the positivity estimates cannot we prove the existence as well as sharp asymptotic estimates for the so called large solutions for any 1 p 2 and point out their main properties we also prove a new local energy inequality for suitable norms of the gradients of the solutions as a consequence we prove that bounded local weak solutions are indeed local strong solutions more precisely tu lloc2

averaging for superconvergence verification and application of 2d edge elements to maxwell s equations in metamaterials abstract in this paper a simple averaging technique for edge elements is justified to improve the standard convergence rate hence achieving superconvergence when they are used to solve the maxwell s equations for simplicity here we focus on the lowest order triangular edge element which is widely used in practice though there exists no natural superconvergence points for the numerical solution obtained on such an edge element superconvergence can be obtained after a simple average of solutions over the neighboring elements a comprehensive analysis for the lowest order triangular edge element is carried out and one order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms formed by two triangles i e superconvergence happens at the parallelogram centers we also provide detailed analysis to explain why several cases do not have superconvergence extensive numerical results consistent with our analysis are presented

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

a spatial high order hexahedral discontinuous galerkin method to solve maxwell s equations in time domain in this paper we present a non dissipative spatial high order discontinuous galerkin method to solve the maxwell equations in the time domain the non intuitive choice of the space of approximation and the basis functions induce an important gain for mass stiffness and jump matrices in terms of memory this spatial approximation combined with a leapfrog scheme in time leads also to a fast explicit and accurate method a study of the dispersive error is carried out and a stability condition for the proposed scheme is established some comparisons with other schemes are presented to validate the new scheme and to point out its advantages finally in order to improve the efficiency of the method in terms of cpu time on general unstructured meshes a strategy of local time stepping is proposed

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

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

a meshless local petrov galerkin method for the time dependent maxwell equations abstract in this paper the meshless local petrov galerkin mlpg method is employed to solve the 2 d time dependent maxwell equations the mlpg method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces in the current work the moving least square reproducing kernel mlsrk scheme is chosen to be the trial function the method is applied for the unsteady maxwell equations in different media in the local weak form by employing the difference operator for evolution in time and simultaneously in time and space the semi discrete and fully discrete schemes are obtained respectively the error estimation is discussed for both the semi discrete and fully discrete numerical schemes for modelling the time dependent maxwell equations we show that provided that the time step size is sufficiently small the proposed scheme yields an error of o 2 m 1 2 in the l 2 norm for the square of error the new scheme is implemented and the numerical results are provided to justify 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