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

a 3d brick element based on hu washizu variational principle for mesh distortion in order to improve the accuracy of the mixed element for irregular meshes a penalty equilibrating 3d mixed element based on the hu washizu variational principle has been proposed in this paper the key idea in this work is to introduce a penalty term into the hu washizu three field functional which can enforce the stress components to satisfy the equilibrium equations in a weak form compared with the classical hybrid and mixed elements this technique can efficiently reduce the sensitivity of the element to mesh distortion the reason for the better results of this penalty technique has been investigated by considering a simple 2d problem from this investigation it has been found that the penalty parameter here plays the role of a scaling factor to reduce the influence of the parasitic strain or stress which is similar to the devised selective scaling factor proposed by sze furthermore compared with the hybrid stress element the proposed element based on the three field variational principle is more suitable for material non linear analysis numerical examples have demonstrated the improved performance of the present element especially in stress computation when fem meshes are irregular copyright 2002 john wiley sons ltd

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

application of triangular space time finite elements to problems of wave propagation abstract the displacement hp space time formulation derived from hamilton s weak principle hwp is studied for simultaneous space time solutions without conventional semi discretizations a special type of discretization of space time is presented in which both space and time dependency are removed the theoretical basis for a displacement hp version space time finite element is developed and a bilinear formulation of space time dynamics is presented in which numerical studies can be performed for h or p version elements of any shape the commitment to a weak formulation of force and momentum allows appropriate solution discontinuities to develop naturally at element boundaries to predict the steep gradients of stress in wave like motions triangular elements are used to cut across space and time such that discontinuities in force or momentum can be located at the element boundary a unified theory of shape functions for space time finite elements is developed and the shape functions for triangular and parallelogram elements are derived in terms of a set of orthogonal functions the assembly procedure for hp version space time finite elements is also discussed to speed the rate of convergence an extraction technique is developed to obtain forces and momenta from integrals of generalized displacements not from their derivatives

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

a pseudospectral method for time domain computation of electromagnetic scattering by bodies of revolution we present a multidomain pseudospectral method for the accurate and efficient time domain computation of scattering by body of revolution bor perfectly electrically conducting objects in the bor formulation of the maxwell equations the azimuthal dependence of the fields is accounted for analytically through a fourier series the numerical scheme in the r z plane is developed in general curvilinear coordinates and the method of characteristics is applied for patching field values in the individual subdomains to obtain the global solution a modified matched layer method is used for terminating the computational domain and special attention is given to proper treatment of the coordinate singularity in the scattered field formulation and correct time domain boundary conditions along edges numerical results for monochromatic plane wave scattering by smooth and nonsmooth axis symmetric objects including spheres cone spheres and finite cylinders is compared with results from the literature illustrating the accuracy and computational efficiency associated with the use of properly constructed spectral methods to emphasize the versatility of the presented framework we compute plane wave scattering by a missile and find satisfactory agreement with method of moment mom computations

error driven dynamical hp meshes with the discontinuous galerkin method for three dimensional wave propagation problems an hp adaptive discontinuous galerkin method for electromagnetic wave propagation phenomena in the time domain is proposed the method is highly efficient and allows for the first time the adaptive full wave simulation of large time dependent problems in three dimensional space refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes for guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed the computational mesh is adapted such that a given error tolerance is respected throughout the entire time domain simulation

the serre spectral sequence of a multiplicative fibration in a fibration omegaf omegaj under right arrow omegax omega pi under right arrow omegab we show that finiteness conditions on f force the homology serre spectral sequence with f p coefficients to collapse at some finite term this in particular implies that as graded vector spaces h omegax is almost isomorphic to h omegab circle times h omegaf one consequence is the conclusion that x is elliptic if and only if b and f are

on an unconditional basis of generalized eigenvectors of the nonself adjoint gribov operator in bargmann space abstract in this article we prove that the system of generalized eigenvectors of the gribov operator is an unconditional basis in bargmann space we also give a generalized diagonalization in the abel sense of the semigroups respectively associated to this operator and to its square root finally some open problems are pointed out

source and boundary implementation in vector and scalar dgtd we summarize the boundary and source implementation for the several formulations of the discontinuous galerkin time domain method dgtd since dgtd with zeroth order scalar basis functions using the upwind flux coincides with the finite volume time domain fvtd many of the concepts developed for fvtd can be ported to dgtd in any of its different formulations scalar vector basis upwind centered flux numerical examples illustrate the different alternatives

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

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

a high order non conforming discontinuous galerkin method for time domain electromagnetics in this paper we discuss the formulation stability and validation of a high order non dissipative discontinuous galerkin dg method for solving maxwell s equations on non conforming simplex meshes the proposed method combines a centered approximation for the numerical fluxes at inter element boundaries with either a second order or a fourth order leap frog time integration scheme moreover the interpolation degree is defined at the element level and the mesh is refined locally in a non conforming way resulting in arbitrary level hanging nodes the method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities numerical experiments with high order elements show the potential of the method

globally constraint preserving fr dg scheme for maxwell s equations at all orders abstract computational electrodynamics ced the numerical solution of maxwell s equations plays an incredibly important role in several problems in science and engineering high accuracy solutions are desired and the discontinuous galerkin dg method is one of the better ways of delivering high accuracy in numerical ced maxwell s equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable balsara and kappeli 2019 presented a von neumann stability analysis of globally constraint preserving dg schemes for ced up to fourth order that paper was focused on developing the theory and documenting the superior dissipation and dispersion of dgtd schemes in media with constant permittivity and permeability in this paper we present working dgtd schemes for ced that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space our dgtd schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh our first finding is that at fourth and higher orders of accuracy one has to evolve some zone centered modes in addition to the face centered modes it is well known that the limiting step in dg schemes causes a reduction of the optimal accuracy of the scheme though the schemes still retain their formal order of accuracy with weno type limiters in this paper we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the dg scheme this very favorable second finding ensures that dgtd schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability we also study the conservation of electromagnetic energy in these problems our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space as long as the conductivity is zero

a hybridizable discontinuous galerkin method combined to a schwarz algorithm for the solution of 3d time harmonic maxwell s equation a schwarz type domain decomposition method is presented for the solution of the system of 3d time harmonic maxwell s equations we introduce a hybridizable discontinuous galerkin hdg scheme for the discretization of the problem based on a tetrahedrization of the computational domain the discrete system of the hdg method on each subdomain is solved by an optimized sparse direct lu factorization solver the solution of the interface system in the domain decomposition framework is accelerated by a krylov subspace method the formulation and the implementation of the resulting dd hdg domain decomposed hybridizable discontinuous galerkin method are detailed numerical results show that the resulting dd hdg solution strategy has an optimal convergence rate and can save both cpu time and memory cost compared to a classical upwind flux based dd dg domain decomposed discontinuous galerkin approach

analyse de scattering d un operateur cubique de heun dans l espace de bargmann the boundary conditions at infinity are used in a description of all maximal dissipative extensions in bargmann space of the minimal heun s operator the characteristic functions of the dissipative extensions are computed completeness theorems are obtained for the system of generalized eigenvectors

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

stable spectral methods on tetrahedral elements a framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented although most of the developments are of a general nature an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron in the first part the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the jacobi polynomials to problems beyond the line this allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and subsequently for the formulation of stable spectral schemes on such unstructured nodal sets the second part of this work is devoted to a discussion of weakly imposed boundary conditions and energy stable schemes are formulated for a wide class of problems exemplified by advection problems advection diffusion problems and linear symmetric hyperbolic systems finally in the third part issues related to computational efficiency and implementation of the schemes are discussed the spectral accuracy of the approximation is confirmed through an example and factorization methods for the efficient computation of derivatives on the general nodal sets within the d simplex are developed ensuring that the proposed schemes are competitive with tensor product based methods in this last part we also show that the advective operator results in an cal o n 2 restriction on the time step similar to that of spectral collocation methods employing a tensor product based approximation the performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain confirming the expected accuracy and stability

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

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

hopf algebras up to homotopy let a d denote a free r reduced differential graded r algebra where r is a commutative ring containing n for 1 a diagonal yi a a 0 a exists which satisfies the hopf algebra axioms including cocommutativity and coassociativity up to homotopy we show that a d must equal u l 3 for some free differential graded lie algebra l 6 this content downloaded from 157 55 39 203 on sat 27 aug 2016 06 23 15 utc all use subject to http about jstor org terms hopf algebras up to homotopy 453 if a is generated as an r algebra in dimensions below rp as a consequence the rational singular chain complex on a topological monoid is seen to be the enveloping algebra of a lie algebra we also deduce for an r connected cw complex x of dimension enveloping algebra and that pth powers vanish in h qx zp department of mathematics massachusetts institute of technology cambridge massachusetts 02139 this content downloaded from 157 55 39 203 on sat 27 aug 2016 06 23 15 utc all use subject to http about jstor org terms

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

a hybrid pstd adi cfdtd method for mixed scale electromagnetic problems we propose a hybrid technique combining the pseudospectral time domain pstd method with the alternating direction implicit conformal finite difference time domain adi cfdtd method to solve 3 d mixed scale problems in computational electromagnetics a mixed scale problem contains both electrically large and relatively homogeneous regions and electrically small fine details thus poses a significant computational challenge to any single computational method if it is utilized alone in particular the adi cfdtd method is an unconditionally stable time domain method with second order spatial accuracy and allows the time step to be increased beyond the courant friedrichs levy limit it is suitable for electrically small problem structure details much smaller than a wavelength but is inefficient and suffers from large numerical errors for electrically large scale regions the pstd method on the other hand is accurate and efficient for regions with large relatively homogeneous materials but loses its efficiency for electrically small structures the hybrid pstd adi cfdtd method overcomes these disadvantages and is potentially more useful than the individual solvers the implementation details and numerical accuracy of this hybrid method are examined numerical examples demonstrate the advantages of the hybrid pstd adi cfdtd method

a short note on the chaoticity of a weight shift on concrete orthonormal basis associated to some fock bargmann space let f c be the hilbert space generated by the orthonormal basis en z 2 1 4e 2z2e 2 n 2 2i n z n n where 0 and are real numbers this space is a particular case of theta fock bargmann spaces recently constructed by ghanmi intissar j math phys 54 063514 2013 in the present work we consider on f c the weight shift operator m defined by men nen 1 n n where n c e2 n and c e 2 and we show the chaoticity of the operator hp mpdp 1 p 0 1 2 where d is adjoint of m

dynamic response and wave propagation in plane trusses and frames the dynamics of planar frames and trusses is analyzed in terms of the propagation of axial longitudinal and flexural transverse stress waves being structural members the waves are multiscattered at the joints and scattering coefficients representing the reflection and transmission of both types of waves at each joint are derived from the dynamics and compatibility conditions of the joint the complex multireflected waves within the structure are evaluated in the frequency domain by a newly developed reverberation matrix which is formulated from scattering coefficients and propagating phase factors transient waves are then analyzed by fourier synthesis and evaluated by a fast fourier transform algorithm transient responses for the axial and bending strains in all structural members are calculated over a long duration for a model truss with rigid joints comparison to experimental data of the model truss under a step loading shows good agreement for the early as well as considerably long time responses

the partition of unity finite element method for short wave acoustic propagation on non uniform potential flows a novel numerical method is proposed for modelling time harmonic acoustic propagation of short wavelength disturbances on non uniform potential flows the method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space the basis functions are local solutions of the governing equations they are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non uniform many wavelengths can be included within a single element leading to ultra sparse meshes results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid based schemes numerical results for lined uniform two dimensional ducts and for non uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved numerical studies indicate that the pollution effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization

optimal energy conserving discontinuous galerkin methods for linear symmetric hyperbolic systems abstract we propose energy conserving discontinuous galerkin dg methods for symmetric linear hyperbolic systems on general unstructured meshes optimal a priori error estimates of order k 1 are obtained for the semi discrete scheme in one dimension and in multi dimensions on cartesian meshes when tensor product polynomials of degree k are used a high order energy conserving lax wendroff time discretization is also presented extensive numerical results in one dimension and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods one particular method with the doubling of unknowns is found to be optimally convergent on triangular meshes for all the examples considered in this paper the method is also compared with the classical dissipative upwinding dg method and conservative dg method with a central flux it is numerically observed for the new method to have a superior performance for long time simulations

regularized trace formula of magic gribov operator on bargmann space abstract in this article we obtain a regularized trace formula for magic gribov operator h g h acting on bargmann space where g a 3 a 3 and h a a i a a a a here a and a are the standard bose annihilation and creation operators and in reggeon field theory the real parameters is the magic coupling of pomeron is pomeron intercept is the triple coupling of pomeron and i 2 1 by applying some abstract results of sadovnichii podol skii 2002 23 we give the number of corrections sufficient for the existence of finite formula of the trace of concrete magic gribov s operator

modelling of short wave diffraction problems using approximating systems of plane waves this paper describes a finite element model for the solution of helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element the approach is based on partition of unity isoparametric elements at each finite element node the potential is expanded in a discrete series of planar waves each propagating at a specified angle these angles can be uniformly distributed or may be carefully chosen they can also be the same for all nodes of the studied mesh or may vary from one node to another the implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres the wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced issues related to the integration and the conditioning are also discussed

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

an implicit hybridized discontinuous galerkin method for the 3d time domain maxwell equations we present a time implicit hybridizable discontinuous galerkin hdg method for numerically solving the system of three dimensional 3d time domain maxwell equations this method can be seen as a fully implicit variant of classical so called dgtd discontinuous galerkin time domain methods that have been extensively studied during the last 10 years for the simulation of time domain electromagnetic wave propagation the proposed method has been implemented for dealing with general 3d problems discretized using unstructured tetrahedral meshes we provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand and its performance when applied to more realistic problems we also include some performance comparisons with a centered flux time implicit dgtd method

stable spectral methods for conservation laws on triangles with unstructured grids this paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle lagrange interpolation on a general two dimensional nodal set is employed and by imposing the boundary conditions weakly through a penalty term the scheme is proven stable in l 2 this result is established for a general unstructured grid in the triangle a special case for which the nodes along the edges of the triangle are chosen as the legendre gauss lobatto quadrature points is discussed in detail the eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an o n 2 restriction on the time step when considering explicit time stepping c 1999 elsevier science s a all rights reserved

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

solution of the time harmonic maxwell equations using discontinuous galerkin methods we present numerical results concerning the solution of the time harmonic maxwell equations discretized by discontinuous galerkin methods in particular a numerical study of the convergence which compares different strategies proposed in the literature for the elliptic maxwell equations is performed in the two dimensional case

on the sensitivity of elastic waves due to structural damages time frequency based indexing method time frequency analysis of various simulated and experimental signals due to elastic wave scattering from damage are performed using wavelet transform wt and hilbert huang transform hht and their performances are compared in context of quantifying the damages spectral finite element method is employed for numerical simulation of wave scattering an analytical study is carried out to study the effects of higher order damage parameters on the reflected wave from a damage based on this study error bounds are computed for the signals in the spectral and also on the time frequency domains it is shown how such an error bound can provide an estimate of error in the modelling of wave propagation in structure with damage measures of damage based on wt and hht is derived to quantify the damage information hidden in the signal the aim of this study is to obtain detailed insights into the problem of 1 identifying localised damages 2 dispersion of multifrequency non stationary signals after they interact with various types of damage and 3 quantifying the damages sensitivity analysis of the signal due to scattered wave based on time frequency representation helps to correlate the variation of damage index measures with respect to the damage parameters like damage size and material degradation factors

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

a sequence of absorbing boundary conditions for maxwell s equations following the scheme developed by engquist and majda math comp 31 1977 629 for first order systems we derive a theoretical perfectly absorbing nonlocal boundary condition for maxwell s equations at a flat outer boundary this condition can be approximated to any desired order by a differential equation on the boundary and a sequence of such equations is developed here in terms of tangential derivatives of the electromagnetic fields at the boundary the resulting set of equations comprising maxwell s equations in the interior together with any of the local boundary conditions is shown to admit no exponentially growing solutions and questions of their well posedness are addressed

adams cobar equivalence let f be the homotopy fibre of a continuous map y omega x with x simply connected we modify and extend a construction of adams to obtain equivalences of dga s and dga modules omega c x congruent to cu omega x and omega c omega y c x congruent to cu f where on the left hand side omega denotes the cobar construction our equivalences are natural in x and omega using this result we show how to read off the algebra h omega x r and the h omega x r module h f r from free models for the singular cochain algebras cs x and cs y here we assume r is a principal ideal domain and x and y are of finite r type

grid stabilization of high order one sided differencing i first order hyperbolic systems we construct stable maximal order boundary closures for high order central difference methods the stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations we find that methods up through 8th order can be stabilized by the addition of a single node up through 16th order by the addition of two nodes and up through 22nd order with three extra nodes we also consider the application of the technique to dispersion relation preserving methods and we construct and test artificial dissipation operators

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

approximation of the semigroup generated by the hamiltonian of reggeon field theory in bargmann space abstract the reggeon field theory is governed by a non self adjoint operator constructed as a polynomial in a a the standard bose annihilation and creation operators in zero transverse dimension this hamiltonian acting in bargmann space is defined by h a 2 a 2 a a i a a a a where i 2 1 and are real numbers and the operators a a satisfy the commutation relation a a i as the quantum mechanical system described by h has a velocity dependent potential containing powers of momentum up to the fourth the problem of existence of hamiltonian path integral for the evolution operator e t h of this theory is of interest on its own in particular can we express e t h as a limit of integral operators in this article one considerably reduces the difficulty by studying the trotter product formula of h to reach two objectives the first objective is to prove a very specific error estimate for the error in a trotter product formula in trace norm for h viewed as the sum of the operators a 2 a 2 and a a i a a a a the second objective of this work is to give a approximation of the semigroup generated by h when h is split in the sum of a 2 a 2 a a and i a a a a we note that this case is entirely different in fact the usual trotter product formula is not defined because the interaction operator a a a a is not the infinitesimal generator of a semigroup on bargmann space for 0 and 0 we choose an approximation operator i i a a a a e a 2 a 2 a a and we give a connection between and e h this choice allows us to give in a intissar note on the path integral formulation of reggeon field theory preprint a generalized trotter product formula for t a a i a a a a i e for limit case as 0 and answers to the above question

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

tetrahedralhpfinite elements we present a new discretisation for the incompressible navier stokes equations that extends spectral methods to three dimensional complex domains consisting of tetrahedral subdomains the algorithm is based on standard concepts ofhpfinite elements as well as tensorial spectral elements this new formulation employs a hierarchical modal basis constructed from a new apex co ordinate system which retains a generalised tensor product these properties enable the development of computationally efficienct algorithms for use on standard finite volume unstructured meshes a detailed analysis is presented that documents the stability and exponential convergence of the method and several flow cases are simulated and compared with analytical and experimental results

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

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

integrating strong and weak discontinuities without integration subcells and example applications in an xfem gfem framework partition of unity methods such as the extended finite element method allows discontinuities to be simulated independently of the mesh int j numer meth engng 1999 45 601 620 this eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re meshing as the discontinuity evolves however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted in this paper we use a simple integration technique proposed for polygonal domains int j nuttier meth engng 2009 80 1 103 134 doi 10 1002 nme 2589 to suppress the need for element subdivision numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi material problem show that the proposed method yields accurate results owing to its simplicity the proposed integration technique can be easily integrated in any existing code copyright c 2010 john wiley sons ltd

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

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

the partition of unity finite element method for elastic wave propagation in reissner mindlin plates this paper reports a numerical method for modelling the elastic wave propagation in plates the method is based on the partition of unity approach in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment the technique is general such that it can be applied to the lagrangian family of finite elements with specific waveform enrichment schemes depending on the dominant modes of wave propagation in the physical system a four noded element for the reissner mindlin plate is derived in this paper which is free of shear locking such a locking free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain as a result the frequency dependent stiffness matrix and the mass matrix are obtained which capture the higher frequency response with even coarse meshes accurately the steps involved in the numerical implementation of such element are discussed in details numerical studies on the performance of the proposed element are reported by considering a number of cases which show very good accuracy and low computational cost copyright 2006 john wiley sons ltd

implicit runge kutta methods and discontinuous galerkin discretizations for linear maxwell s equations in this paper we consider implicit runge kutta methods for the time integration of linear maxwell s equations we first present error bounds for the abstract cauchy problem which respect the unboundedness of the differential operators using energy techniques the error bounds hold for algebraically stable and coercive methods such as gauss and radau collocation methods the results for the abstract evolution equation are then combined with a discontinuous galerkin discretization in space using upwind fluxes for the case that permeability and permittivity are piecewise constant functions we show error bounds for the full discretization where the constants do not deteriorate if the spatial mesh width tends to zero

high order non conforming multi element discontinuous galerkin method for time domain electromagnetics we study a high order discontinuous galerkin time domain dgtd method for solving the system of maxwell equations this method is formulated on a non conforming multi element mesh mixing an unstructured triangulation for the discretization of irregularly shaped objects with a structured cartesian quadrangulation for the rest of the computational domain within each element the electromagnetic field components are approximated by a high order nodal polynomial the dg method proposed here makes use of a centered scheme for the definition of the numerical traces of the electric and magnetic fields at element interfaces associated to a second order or fourth order leap frog scheme for the time integration of the associated semi discrete equations we formulate this dgtd method in 3d and study its theoretical properties in particular an a priori convergence estimation is elaborated finally we present numerical results for the application of the method to 2d test problems

a propos de conjectures de serre et sullivan we first give a new proof of a conjecture of j p serre on the homotopy of finite complexes which was recently proved by c mcgibbon and j neisendorfer the emphasis is on a property of the mod 2 homology of certain spaces their quasi boundedness as right modules over the steenrod algebra this property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anh space then we show that this notion of quasi boundedness simplifies h miller s proof of d sullivan s conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe 2 into a finite complex

an accurate spectral discontinuous finite element formulation of a phase space based level set approach to geometrical optics in this paper we introduce a new numerical procedure for simulations in geometrical optics that based on the recent development of eulerian phase space formulations of the model can deliver very accurate uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations following previous treatments the scheme is based on the evolution of a wavefront in phase space defined as the intersection of level sets satisfying the relevant liouville equation in contrast with previous work however our numerical approximation is specifically designed i to take full advantage of the smoothness of solutions ii to facilitate the treatment of scattering obstacles all while retaining high order convergence characteristics indeed to incorporate the full regularity of solutions that results from the unfolding of singularities our method is based on their spectral representation to enable a simple high order treatment of scattering boundaries on the other hand we resort to a discontinuous galerkin finite element method for the solution of the resulting system of equations the procedure is complemented with the use of a recently derived strong stability preserving runge kutta ssp rk scheme for the time integration that as we demonstrate allows for overall approximations that are rapidly convergent

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

calculation of the sar induced in head tissues using a high order dgtd method and triangulated geometrical models the great majority of numerical calculations of the specific absorption rate sar induced in human tissues exposed to microwaves are performed using the finite difference time domain fdtd method and voxel based geometrical models the straightforward implementation of the method and its computational efficiency are among the main reasons for fdtd being currently the leading method for numerical assessment of human exposure to electromagnetic waves however the rather difficult departure from the commonly used cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context in particular interfaces between tissues where sharp gradients of the electromagnetic field may occur are hardly modeled rigorously in these studies we present here an alternative numerical dosimetry methodology which is based on a high order discontinuous galerkin time domain dgtd method and adapted geometrical models constructed from unstructured triangulations of tissue interfaces and discuss its application to the calculation of the sar induced in head tissues

non dissipative space time hp discontinuous galerkin method for the time dependent maxwell equations a finite element method for the solution of the time dependent maxwell equations in mixed form is presented the method allows for local hp refinement in space and in time to this end a space time galerkin approach is employed in contrast to the space time dg method introduced in 1 test and trial spaces do not coincide this allows for obtaining a non dissipative method to obtain an efficient implementation a hierarchical tensor product basis in space and time is proposed this allows to evaluate the local residual with a complexity of o p 4 and o p 5 for affine and non affine elements respectively