a bloch decomposition based stochastic galerkin method for quantum dynamics with a random external potential in this paper we consider the numerical solution of the one dimensional schrodinger equation with a periodic lattice potential and a random external potential this is an important model in solid state physics where the randomness results from complicated phenomena that are not exactly known here we generalize the bloch decomposition based time splitting pseudospectral method to the stochastic setting using the generalized polynomial chaos with a galerkin procedure so that the main effects of dispersion and periodic potential are still computed together we prove that our method is unconditionally stable and numerical examples show that it has other nice properties and is more efficient than the traditional method finally we give some numerical evidence for the well known phenomenon of anderson localization

decorrelation estimates for random discrete schrodinger operators in dimension one and applications to spectral statistics the purpose of the present work is to establish decorrelation estimates for some random discrete schrodinger operator in dimension one we prove that the minami estimates are consequences of the wegner estimates and localization we also prove decorrelation estimates at distinct energies for the random hopping model and schrodinger operators with alloy type potentials these results are used to give a description of the spectral statistics

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

a time space flux corrected transport finite element formulation for solving multi dimensional advection diffusion reaction equations abstract we present a time space flux corrected transport fct finite element formulation for the multi dimensional time dependent advection diffusion reaction equation monotonic solutions can be achieved with the presented method while large time steps with courant number c r 1 are used numerical verification as well as a grid convergence analysis are carried out for 1d and 2d benchmark problems a 1d burgers equation with various reynolds numbers is solved with the time space fct method in addition a 2d transport problem with and without a nonlinear reaction inside of a cavity is considered finally the newly developed time space fct formulation is applied for modeling biofilm growth problems based on a continuum mathematical model it turns out that the time space fct method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time space finite incremental calculus fic method

a unified set of single step asymptotic annihilation algorithms for structural dynamics when solving the equations of structural dynamics using direct time integration meth6ds algorithmic damping is useful in controlling spurious high frequency oscillations ideally an algorithm should possess asymptotic annihilation of the highfrequency response i e spurious oscillations in the high frequencies are eliminated after one time step in this paper a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low frequency regime the algorithms are based upon using finite elements in time the displacement and velocity fields are interpolated independently using time discontinuous functions the equations of motion displacement velocity compatibility and the time continuity of displacement and velocity are satisfied weakly asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time an analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the pad6 approximation table results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime

time discontinuous stabilized space time finite elements for timoshenko beams stabilized space time e nite element methods for the transient computational analysis of the elastodynamics of timoshenko beams have been developed the underlying time discontinuous galerkin formulation is implicit unconditionally stable higher order accurate and robust the employed interpolations are continuous in space and time inside time slabs but discontinuous in time between adjacent time slabs to suppress spurious numerical oscillations near discontinuities or high gradients galerkin least squares stabilization has been applied further improvement of the numerical representation of the transient elastic wave propagation phenomena has been achieved by specially designed galerkin gradient least squares operators because of the reduced phase and amplitude errors of these e nite elements much coarser spatial meshes may be used without loss of accuracy the resulting reduction of the number of unknowns leads to a signie cant decrease of computer time furthermore a globaladaptivetime stepping strategy has beenemployed basedon thetemporal jump residual of thetimee niteelement method as error estimator numerical examples of transientelastic wave propagation in timoshenko beams involving a wide spectrum of wave numbers and frequencies demonstrate the good performance of the developed methods

extended framework of hamilton s principle applied to duffing oscillation abstract the paper begins with a novel variational formulation of duffing equation using the extended framework of hamilton s principle ehp unlike the original hamilton s principle this formulation properly accounts for initial conditions and it recovers all the governing differential equations as its euler lagrange equation thus it serves base for the development of novel computational methods involving finite element representation over time for its feasibility we provide the simplest temporal finite element method by adopting linear temporal shape functions numerical examples are included to verify and investigate performance of non iterative algorithm in the developed method

higher order accurate time step integration algorithms by post integration techniques in this paper a framework to construct higher order accurate time step integration algorithms based on the post integration techniques is presented the prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced the algorithmic parameters are chosen such that unconditionally a stable higher order accurate time step integration algorithms with controllable numerical dissipation can be constructed for linear problems besides it is shown that the order of accuracy for non linear problems is maintained through the relationship between the present formulation and the runge kutta method the second order differential equations are also considered numerical examples are given to illustrate the validity of the present formulation copyright 2001 john wiley sons ltd

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

a simple em algorithm for capture recapture data with categorical covariates a simple em algorithm is proposed for obtaining maximum likelihood estimates when fitting a loglinear model to data from k capture recapture samples with categorical covariates the method is used to analyze data on screening for the early detection of breast cancer

internal lifschitz singularities for one dimensional schrodinger operators the integrated density of states of the periodic plus random one dimensional schrodinger operator h omega delta v per sum limits i qi left omega right f left o i right f 0 q i 0 has lifschitz singularities at the edges of the gaps insp h we use dirichlet neumann bracketing based on a specifically one dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones

consistent discontinuous finite elements in elastodynamics finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non smooth problems in this work we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems we first review some formulations for the elastostatic problem and prove two korn like inequalities which are very useful for the derivation of convergence rates in sobolev norms next we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions followed by numerical investigations of a typically non smooth problem involving classical and emerging variational formulations we also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise lagrangian polynomials

perturbation of resonances in quantum mechanics schrodinger operators on l2 r3 of the form v with potentials v real analytic in are discussed the analytic structure in v and k with k2 the energy variable of the resolvent kernel the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions the lower order expansion coefficients are computed explicitly the resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions this gives asymptotic low energy information about schrodinger hamiltonians with short range potentials the perturbation theory of resonances eigenvalues and of the corresponding functions for hamiltonians describing n point interactions perturbed by a potential is also given

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

asymptotic behavior of the integrated density of states of acoustic operators with random long range perturbations in this paper we study the behavior of the integrated density of states of random acoustic operators of the form a 1 when is considered as an anderson type long range perturbations of some periodic function the behavior of the integrated density of states of a in the vicinity of the internal spectral edges is given

on the integrated density of states for crystals with randomly distributed impurities in the present paper we discuss spectral properties of a periodic schrodinger operator which is perturbed by randomly distributed impurities such operators occur as simple models for crystals or semi conductors with impurities while the spectrum itself is independent of the concentrationp of impurities for 0 p 1 we focus our attention on the limiting behavior of the integrated density of states p of the random schrodinger operator inside a spectral gap of the periodic operator asp 0 denoting byu0 the set of eigenvalues in the gap of the reference problem having precisely one impurity located at the origin say we show that the integrated density of states concentrates around the points ofu0 in the sense that p ue is of orderp for any fixed e neighborhoodue ofu0 while p k c p2 for any compact subsetk of the gap which does not intersectue

a discontinuous galerkin finite element method for dynamic and wave propagation problems in non linear solids and saturated porous media a time discontinuous galerkin finite element method dgfem for dynamics and wave propagation in non linear solids and saturated porous media is presented the main distinct characteristic of the proposed dgfem is that the specific p3 p1 interpolation approximation which uses piecewise cubic hermite s polynomial and linear interpolations for both displacements and velocities in the time domain is particularly proposed consequently continuity of the displacement vector at each discrete time instant is exactly ensured whereas discontinuity of the velocity vector at the discrete time levels still remains the computational cost is then obviously saved particularly in the materially non linear problems as compared with that required for the existing dgfem both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non linear problems numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional galerkin finite element method using the newmark algorithm in the time domain copyright 2003 john wiley sons ltd

extended framework of hamiltons principle for thermoelastic continua based upon the extended framework of hamiltons principle a variational formulation for fully coupled thermoelasticity is presented the resulting formulation can properly account for all the governing differential equations as well as initial boundary conditions thus it provides the basis for a class of unified spacetime finite element methods by employing bar elements in one dimensional space along with linear shape functions temporally the simplest spacetime finite element method is presented herein with representative examples for its validity

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

internal lifshits tails for random magnetic schrodinger operators this paper is devoted to the study of lifshits tails for weak random magnetic perturbations of periodic schrodinger operators acting on l2 rd of the form h w i zdw a 2 v where v is a zd periodic potential is positive coupling constants w zd are i i d and bounded random variables and a c01 rd rd is the single site vector magnetic potential we prove that for small at an open band edge a true lifshits tail for the random magnetic schrodinger operator occurs if a certain set of conditions on h0 v and on a holds

inadmissibility of the maximum likelihood estimator in the sequential estimation of the size of a population summary for the sequential estimation of the size of a finite population the maximum likelihood estimator is a commonly used estimator but has no explicit expression in this paper it is shown that the maximum likelihood estimator is inadmissible an estimator which is simple in form and dominates the maximum likelihood estimator asymptotically is proposed to study the optimality property of the estimator for the stopping rule sb from a decision theoretical approach we investigate the admissibility of the estimator under a general weighted squared error loss function although the maximum likelihood estimator and its risk have no closed form expression the estimator will be shown to be inadmissible by means of a general theorem that we establish later as theorem 2 1 in this paper we only consider the simple capture capture model discussed by samuel 1968 more complex and realistic capture recapture models have been developed subsequently pollack otto 1983 pickands raghavachari 1987 bonett woodward bentler 1987 cormack 1989 huggins 1989 we conjecture that many of the

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

the spectrum minimum for random schrodinger operators with indefinite sign potentials this paper sets out to study the spectral minimum for an operator belonging to the family of random schrodinger operators of the form h wper v where we suppose that v is of anderson type and the single site is assumed to be with an indefinite sign under some assumptions we prove that there exists 0 0 such that for any 0 0 the minimum of the spectrum of h is obtained by a given realization of the random variables

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

expansion of the almost sure spectrum in the weak disorder regime abstract the spectrum of random ergodic schrodinger type operators is almost surely a deterministic subset of the real line the random operator can be considered as a perturbation of a periodic one as soon as the disorder is switched on via a global coupling constant the spectrum expands we estimate how much the spectrum expands at its bottom for operators on l 2 z d

support theorems for random schrodinger operators in the multi dimensional case it is shown that the increase of the topological support of the probability measure describing the randomness of potentials implies the increase of the spectrum in the one dimensional case the converse statement for the absolutely continuous spectrum is valid especially the spectrum in general dimension and the absolutely continuous spectrum in one dimension are determined only by the topological support of the random potentials

an efficient time discontinuous galerkin procedure for non linear structural dynamics this paper presents an effective time discontinuous galerkin procedure for non linear dynamics the procedure is based on a cheap iterative algorithm which is rather different from previous efforts in fact the algorithm is designed so that the corrected solutions inherit the desired stability and dissipative properties and the iterations serve only to improve accuracy some numerical tests illustrate the good performance of the present procedure which appears to be competitive with the available implementations of time discontinuous galerkin methods

time discontinuous galerkin methods with energy decaying correction for non linear elastodynamics in this paper a new time discontinuous galerkin tdg formulation for non linear elastodynamics is presented the new formulation embeds an energy correction which ensures truly energy decaying thus allowing to achieve unconditional stability that as shown in the paper is not guaranteed by the classical tdg formulation the resulting method is simple and easily implementable into existing finite element codes moreover it inherits the desirable higher order accuracy and high frequency dissipation properties of the classical formulation numerical results illustrate the very good performance of the proposed formulation copyright 2010 john wiley sons ltd

estimating the size of closed populations using inverse multiple recapture sampling a log linear model for estimating the size of a closed population is defined for inverse multiple recapture sampling with dependent samples efficient estimators of the log linear model parameters and the population size are obtained by the method of minimum chi square a chi square test of the general linear hypothesis regarding the log linear model parameters is defined

large deviations and lifshitz singularity of the integrated density of states of random hamiltonians we consider the integrated density of states ids of random hamiltonian h v v being a random field on d which satisfies a mixing condition we prove that the probability of large fluctuations of the finite volume ids 1 h d around the thermodynamic limit is bounded from above by exp k k 0 in this case can be recovered from a variational principle furthermore we show the existence of a lifshitztype of singularity of as 0 in the case where v is non negative more precisely we prove the following bound exp k d 2 as 0 k 0 this last result is then discussed in some examples

time discontinuous material point method for transient problems abstract this paper presents a time discontinuous material point method tdmpm for transient problems such as the wave propagation and impact responses in solids by dividing the continuous time domain into discrete time intervals the weak form of the tdmpm is established by considering the discrete grid based governing equations constraint and discontinuity conditions the displacement and velocity fields in a time interval are interpolated with the piecewise cubic and linear functions respectively by substituting the assumed displacement and velocity fields into the weak form a novel computational framework for the grid displacements and velocities at the discrete time instants is constructed in the new formulations the displacement field at each time instant remains to be continuous whereas the velocity field at the time instant becomes discontinuous these unique features ensure the tdmpm could properly capture the discontinuous characteristics and control the spurious numerical oscillations two numerical examples under the impact loading are used to verify the proposed method two representative impact problems are then presented for further verification and demonstration besides the corresponding contact algorithm adopted in the tdmpm is shown to be capable of capturing the correct contact behavior with higher fidelity and less computational cost than the mpm the presented results illustrate that the tdmpm could successfully control the spurious numerical oscillations associated with transient simulations

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

2 dimensional localization of acoustic waves in random perturbation of periodic media abstract in this work we study internal band edges localization of acoustic waves in 2 dimensional space obtained by random perturbation of some periodic media our results rely on the study of lifshitz tails for the integrated density of states for random acoustic operators of the form a 1 localization is then deduced by the standard multiscale argument

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

adaptive finite element procedures for linear and non linear dynamics this paper discusses implementation and adaptivity of the discontinuous galerkin dg finite element method as applied to linear and non linear structural dynamic problems by the dg method both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived they are third order accurate and while the implicit procedure is unconditionally stable the explicit one is conditionally stable an h adaptive procedure based on the zienkiewicz zhu error estimate using the spr technique is applied numerical examples are presented to show the suitability of the dg method for both linear and non linear structural dynamic analysis copyright 1999 john wiley sons ltd

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

on energy gaps in a new type of analytically solvable model in quantum mechanics abstract a new type of analytically solvable one dimensional model in quantum mechanics formally given by d2 dx2 j z j x yj j yj r is analyzed besides calculating the density of states of various periodic arrangements of such and models we also prove a generalization of the saxon hutner conjecture concerning the existence of gaps in the energy spectrum of certain substitutional alloys of or interactions in addition the spectrum of the corresponding random hamiltonians i e j j z ergodic stochastic processes is investigated and explicitly determined in certain cases

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

a low concentration asymptotic expansion for the density of states of a random schrodinger operator with poisson disorder abstract in this paper we study the density of states of a random schrodinger operator of the formh v wherev is a poisson potential i e a poisson random field of concentration we show thatn d the density of states ofh admits an asymptotic expansion in when 0 then we use this expansion to deduce the behaviour of the integrated density of states ofh in the energy interval 0 when goes to 0 dans ce travail nous etudions la densite d etats d un operateur de schrodinger aleatoire de la formeh v ouv est un potentiel de poisson i e un champ aleatoire de poisson de concentration nous demontrons l existence d un developpement asymptotique en pourn d la densite d etat deh ceci quand 0 puis nous utilisons ce developpement pour estimer la taille de la densite d etats integree deh dans l intervalle d energie 0 quand tend vers 0

extended framework of hamilton s principle for continuum dynamics abstract hamilton s principle is the variational principle for dynamical systems and it has been widely used in mathematical physics and engineering however it has a critical weakness termed end point constraints which means that in the weak form we cannot use the given initial conditions properly by utilizing a mixed formulation and sequentially assigning initial conditions this paper presents a novel extended framework of hamilton s principle for continuum dynamics to resolve such weakness the primary applications lie in an elastic and a j 2 viscoplastic continuum dynamics the framework is simple and initiates the development of a space time finite element method with the proper use of initial conditions non iterative numerical algorithms for both elasticity and j 2 viscoplasticity are presented

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

brownian survival and lifshitz tail in perturbed lattice disorder abstract we consider the annealed asymptotics for the survival probability of brownian motion among randomly distributed traps the configuration of the traps is given by independent displacements of the lattice points we determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant as applications we show the lifshitz tail effect of the density of states of the associated random schrodinger operator and derive a quantitative estimate for the strength of intermittency in the parabolic anderson problem

space time topology optimization for one dimensional wave propagation abstract a space time extension of the topology optimization method is presented the formulation with design variables in both the spatial and temporal domains is used to create structures with an optimized distribution of material properties that can vary in time the method is outlined for one dimensional transient wave propagation in an elastic rod with time dependent young s modulus by two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber frequency shifts and lack of energy conservation the optimization method s potential for creating structures with novel dynamic behavior is illustrated by a simple example it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure optimized designs in form of spatio temporal laminates and checkerboards are generated and discussed the example lays the foundation for creating designs with more advanced functionalities in future work

numerical investigation of constrained direct solutions using hamilton s law this paper presents numerical results obtained for a dynamic system that is modeled using hamilton s law of varying action and for which geometric constraints are enforced explicitly these results are compared with numerical results for the same system that were obtained from ordinary differential equations differential algebraic equations and a formulation based on hamilton s law in which the constraints were enforced implicitly this comparison had two objectives first to show that numerical results from the formulation based on hamilton s law and explicit constraints were equal or superior to results from other formulations second to verify that explicit constraints must be enforced at the end of the time step both objectives were successfully accomplished

weighting parameters for time step integration algorithms with predetermined coefficients in this paper the effect of using the predetermined coefficients in constructing time step integration algorithms is investigated both first and second order equations are considered the approximate solution is assumed to be in a form of polynomial in the time domain it can be related to the truncated taylor s series expansion of the exact solution therefore some of the coefficients can be predetermined from the known initial conditions if there are m predetermined coefficients and r unknown coefficients in the approximate solution the unknowns can be solved by the weighted residual method the weighting parameter method is used to investigate the resultant algorithm characteristics it is shown that the formulation is consistent with a minimum order of accuracy m r the maximum order of accuracy achievable is m 2r unconditionally stable algorithms exist if m r for first order equations and m 1 r for second order equations hence the dahlquist s theorem is generalized algorithms equivalent to the pade approximations and unconditionally stable algorithms equivalent to the generalized pade approximations are constructed the corresponding weighting parameters and weighting functions for the pade and generalized pade approximations are given explicitly copyright 2000 john wiley sons ltd

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

mixed variational principles for dynamic response of thermoelastic and poroelastic continua abstract the primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other this begins with the development of a new stationary principle involving mixed field variables for continuum problems in infinitesimal dynamic thermoelasticity by defining lagrangian and dissipation functions in terms of physically relevant contributions and invoking the rayleigh formalism for damped systems we are able to recover the governing equations of thermoelasticity as the euler lagrange equations this includes the balance laws of linear momentum and entropy energy the constitutive models for elastic response and heat conduction and the natural boundary conditions by including energy contributions associated with second sound phenomena one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry which is most easily seen in the fourier wave number domain a related formulation for dynamic poroelasticity yields two new stationary mixed variational principles depending upon the selection of primary field variables these governing equations can also exhibit an elegant structure which can deepen our understanding of the underlying phenomena and the thermoelastic poroelastic analogy in addition to the theoretical significance the variational formulations developed here can provide the basis for a class of optimization based methods for computational mechanics

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

a particular integral bem time discontinuous fem methodology for solving 2 d elastodynamic problems abstract this study proposes a time discontinuous galerkin finite element method fem for solving second order ordinary differential equations in the time domain the equations are formulated using a particular integral boundary element method bem in the space domain for elastodynamic problems the particular integral bem technique depends only on elastostatic displacement and traction fundamental solutions without resorting to commonly used complex fundamental solutions for elastodynamic problems based on the time discontinuous galerkin fem the unknown displacements and velocities are approximated as piecewise linear functions in the time domain and are permitted to be discontinuous at the discrete time levels this leads to stable and third order accurate solution algorithms for ordinary differential equations numerical results using the time discontinuous galerkin fem are compared with results using a conventional finite difference method the houbolt method both methods are employed for a particular integral bem analysis in elastodynamics this comparison reveals that the time discontinuous galerkin fem is more stable and more accurate than the traditional finite difference methods

random hamiltonians ergodic in all but one direction letv 1 andv 2 be two ergodic random potentials on d we consider the schrodinger operatorh h0 v withh0 and forx x1 x2 v omega x left begin array 20 c v omega 1 x if x 1 over smash scriptstyle vphantom x 0 end array right we prove certain ergodic properties of the spectrum for this model and express the integrated density of states in terms of the density of states of the stationary potentialsv 1 andv 2 finally we prove the existence of the density of surface states forh

formulation and analysis of variational methods for time integration of linear elastodynamics abstract a general framework for variational formulations of linear elastodynamics is introduced variational formulations which originate from the principle of minimum potential energy the hellinger reissner principle and the hu washizu principle are considered they are characterized by the weak enforcement of the velocity displacement relation and of the initial displacement and velocity special cases are obtained by a priori enforcement of these conditions the time integration algorithms obtained discretizing in the space and time domains are studied with reference to accuracy stability and high frequency behaviour it is shown that some of these algorithms are very competitive with respect to classical finite difference methods the only drawback of variational methods seems to be the large size of the set of simultaneous equations which they generate effective solution procedures aiming at minimizing both the storage and the computational effort will be discussed in a separate paper

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

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

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

localization for anderson models on metric and discrete tree graphs we establish spectral and dynamical localization for several anderson models on metric and discrete radial trees the localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies all results are proved under the minimal hypothesis on the type of disorder the random variables generating the trees assume at least two distinct values this level of generality in particular allows us to treat radial trees with disordered geometry as well as schrodinger operators with bernoulli type singular potentials our methods are based on an interplay between graph theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half line

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

global bounds for the lyapunov exponent and the integrated density of states of random schrodinger operators in one dimension in this paper we prove an upper bound for the lyapunov exponent e and a two sided bound for the integrated density of states n e at an arbitrary energy e 0 of random schrodinger operators in one dimension these schrodinger operators are given by potentials of identical shape centred at every lattice site but with non overlapping supports and with randomly varying coupling constants both types of bounds only involve scattering data for the single site potential they show in particular that both e and n e e 1 2 decay at infinity at least like 1 e 1 2 as an example we consider the random kronig penney model

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability