spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

nonlinear dynamics in driven damped sine gordon systems abstract many quasi one and two dimensional systems can be modeled by sine gordon equations illustrations are given for driven and damped sine gordon equations of behaviors that are typical of ways that complexity in time and space can be manifested examples include an ac response to a dc driver and spontaneous pattern formation and low dimensional chaos under the influence of ac or ac dc drivers the importance of the coexistence of coherence and chaos for transport phenomena is emphasized

adaptive numerical analysis in primal elastoplasticity with hardening the quasi static viscoplastic resp elastoplastic evolution problem with isotropic or kinematic hardening is considered with emphasis on optimal convergence and adapted mesh refining in the spatial discretization within one time step of an implicit time discretization the finite element method leads to a minimisation problem for non smooth convex functions on discrete subspaces for piecewise constant resp affine ansatz functions the stress resp displacement approximations are experimentally and theoretically shown to converge linearly an a posteriori error estimate then justifies an automatic adaptive mesh refining algorithm numerical examples support the superiority of the adapted mesh

reliable and efficient equilibrated a posteriori finite element error control in elastoplasticity and elastoviscoplasticity with hardening this paper establishes the reliability and efficiency of equilibrated a posteriori estimates for l 2 stress error control of conforming displacement finite element approximations of incremental plasticity and viscoplasticity with hardening explicit expressions for upper bounds of the reliability constant that enters the guaranteed upper error bound illustrate its crucial dependence on the hardening material constants numerical experiments show that adaptive finite element solutions with marking strategy based on the max refinement rule and local equilibrated error estimators lead to optimal empirical convergence rates 2005 elsevier b v all rights reserved

a quick perturbative method for schrodinger equations within the framework of perturbation theory we propose firstly an iterative method which may serve as a source of optimal unperturbed solutions in both one and more dimensions it combines the runge kutta and newton algorithm and its efficiency is illustrated on a few quartic oscillators secondly admitting also an arbitrary perturbation of potentials we generalize the existing runge kutta one dimensional rayleigh schrodinger constructions of energies and wavefunctions to more dimensions

computational procedure for sturm liouville problems the approaches of datzeff c r acad bulg sci 12 1959 113 285 and wittrick and williams quart j mech appl math xxiv 1971 263 int j mech sci 16 1974 209 have been incorporated to develop a procedure for the automatic computation of the eigen values and the eigen functions of one dimensional linear sturm liouville boundary value eigenproblems both singular and nonsingular the continuous coefficients of a regular sturm liouville problem have been approximated by a finite number of step functions in each step the resulting boundary value problem has been integrated exactly and the solutions have then been matched to construct the continuously differentiable solution of the original problem and the corresponding eigencondition step sizes have been chosen automatically so that the local error has been held in a predetermined interval representative test examples have been computed to illustrate the accuracy reliability and efficiency of the algorithm proposed

averaging technique for fe a posteriori error control in elasticity part i conforming fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper the reliability of any averaging estimator is shown for low order finite element methods in elasticity theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides and independent of the structure of a shape regular mesh

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part ii higher order fem averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing in the second paper of our analysis of their reliability we consider conforming h fem of higher i e not of lowest order in two or three space dimensions in this paper reliablility is shown for conforming higher order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of local averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

nonlinear fourier analysis for the infinite interval korteweg de vries equation i an algorithm for the direct scattering transform the nonlinear fourier analysis of wave motion governed approximately by the korteweg de vries kdv equation on the infinite line is the central point of discussion we assume that the wave amplitude is recorded in the form of a discrete space or time series which is determined either by experimental measurement or by computer simulation of the physical system of interest we develop numerical data analysis procedures based upon the scattering transform solution to the kdv equation as given by gardner et al 1 we are motivated by the observation that historically the fourier transform has been ubiquitously used to spectrally analyze linear wave data here we develop methods for employing the scattering transform as a tool to similarly analyze nonlinear wave data specifically we develop numerical methods to evaluate the direct scattering transform dst of a space or time series the approach thus provides a basis for analyzing and interpreting nonlinear wave behavior in the wavenumber or frequency domain the dst spectrum separates naturally into soliton and radiation components and may be simply interpreted in terms of the large time asymptotic state of the infinite line kdv equation

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

averaging technique for fe a posteriori error control in elasticity part ii λ independent estimates in the second part of our investigation on a posteriori error estimates and a posteriori error control in finite element analysis in elasticity we focus on robust a posteriori error bounds first we establish a residual based a posteriori error estimate which is reliable and efficient up to higher order terms and independent multiplicative constants the lame constant steers the incompressibility second we show the robust efficiency and reliability of averaging techniques in certain norms numerical evidence supports that the reliability of depends on the smoothness of given right hand sides and is independent of the structure of a shape regular mesh

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

averaging techniques for reliable a posteriori fe error control in elastoplasticity with hardening averaging techniques are popular tools in adaptive finite element methods for numerical simulation in continuum mechanics since they provide efficient a posteriori error control in this paper the reliability of any averaging estimator is shown for low order finite element methods in one time step of elastoplasticity with hardening the constants and higher order terms are effected by the hardening and the smoothness of given right hand sides but are independent of the structure of a shape regular mesh since it involves a different functional analytical framework the case of perfect plasticity is excluded from this paper

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable the recently proposed method of calculating perturbation energies using a non normalizable wavefunction by skala and cizek is analysed and rigorously proved

on the numerical integration of the schrodinger equation in the finite difference schemes abstract formulae for the integration of the schrodinger equation for bound states based on the scheme of central differences are generated from the taylor expansion with the help of formal pade approximants these methods are studied in matrix form and a limiting formula the best for a given discretization is obtained

the solitons of zabusky and kruskal revisited perspective in terms of the periodic spectral transform abstract we consider the numerical soliton experiments of zabusky and kruskal 1 and address their results in light of the spectral structure of the periodic korteweg devries kdv equation we derive and exploit a simple relationship between the infinite line transfer matrix and the monodromy matrix of periodic theory this establishes the connection at all stages of our analysis between the infinite line soliton and the soliton limit of periodic theory we develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic fourier analysis finally we introduce the concept of a reference level which is defined to be the level upon which the solitons propagate these results together with judicious numerical experiments allow us to reassess the work of zabusky and kruskal from the standpoint of periodic theory for the kdv equation we find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper we also find that even in the presence of periodic boundary conditions the solitons behave like infinite line solitons in that they undergo phase shifts upon collision with each other our estimate of the fpu recurrence time which includes the effect of phase shifting is within one percent of their observed value

numerical analysis of time depending primal elastoplasticity with hardening the quasi static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme in each time step of a generalized midpoint scheme such as the implicit euler or the crank nicolson scheme the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear resp piecewise constant trial functions an a priori error estimate is established for the fully discrete method which for smooth data and a smooth exact solution proves linear convergence as the mesh size tends to zero strong convergence of the time derivatives is established under mild conditions on the mesh and time step sizes numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the crank nicolson scheme is not always superior over the implicit euler scheme in practice

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

anharmonic oscillator constructing the strong coupling expansions two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential v x 1 2 x2 g 4 x4 are proposed the first one is simply a straightforward solution of the schrodinger equation via the nonlinearization technique resulting in the rapidly convergent perturbation series the second one is a version of the path integral perturbation theory but with an unconventional choice of the zeroth approximation action nine leading coefficients of the strong coupling expansion are computed they decrease rapidly the ninth one being of the order of 10 9 three leading corrections of the nonlinearization approach provide the ground state energy within a relative accuracy of 10 7 10 9 at an arbitrary coupling g the explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g in the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak

fine structure of the asymptotic expansion of cyclic integrals the asymptotic expansion of n dimensional cyclic integrals was expressed as a series of functionals acting on the symmetric function involved in the cyclic integral in this article we give an explicit formula for the action of these functionals on a specific class of symmetric functions these results are necessary for the computation of the o 1 part in the long distance asymptotic behavior of correlation functions in integrable models

adaptive fem analysis of selected elastic visco plastic problems abstract elastic visco plastic deformations modeled by the bodner partom constitutive equations s r bodner y partom constitutive equations elastic viscoplastic strain hardening materials j appl mech 42 1975 385 389 have been evaluated by the adaptive finite element method j m bass j t oden adaptive finite element methods for a class of evolution problems in viscoplasticity int j engrg sci 25 623 653 1987 l demkowicz j t oden w rachowicz o hardy toward a universal hp adaptive finite element strategy part 1 constrained approximation and data structure comput methods appl mech engrg 77 1989 79 112 we use the explicit residual error estimate i babuska w c rheinboldt error estimates for adaptive finite element computations int j numer methods engrg 12 1978 1597 1615 supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the gauss integration points we also address the problems of adaptation strategy and the method of solution transfer to a new mesh results of recent tests confirm efficiency and robustness of the approach

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

determinant representation for dynamical correlation functions of the quantum nonlinear schrodinger equation painleve analysis of correlation functions of the impenetrable bose gas by m jimbo t miwa y mori and m sato 1 was based on the determinant representation of these correlation functions obtained by a lenard 2 the impenetrable bose gas is the free fermionic case of the quantum nonlinear schrodinger equation in this paper we generalize the lenard determinant representation for langle psi 0 0 psi dagger x t rangle to the non free fermionic case we also include time and temeprature dependence in forthcoming publications we shall perform the jmms analysis of this correlationl function this will give us a completely integrable equation and asymptotic for the quantum correlation function of interacting fermions

a convergent adaptive finite element method for the primal problem of elastoplasticity summary the boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional this paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and up to higher order terms the r linear convergence of the stresses with respect to the number of loops applications include several plasticity models linear isotropic kinematic hardening linear kinematic hardening and multisurface plasticity as model for nonlinear hardening laws for perfect plasticity the adaptive algorithm yields strong convergence of the stresses numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule copyright q 2006 john wiley sons ltd

a posteriori error estimation and mesh adaptation for finite element models in elasto plasticity a new approach to a posteriori error estimation and adaptive mesh design based on techniques from optimal control is presented for primal mixed finite element models in elasto plasticity this method uses global duality arguments for deriving weighted a posteriori error bounds for arbitrary functionals of the error representing physical quantities of interest in these estimates local residuals of the computed solution are multiplied by certain weights which are obtained by solving a linearized global dual problem numerically the resulting local error indicators are used in a feed back process for generating economical meshes this approach is developed here for the hencky and prandtl reuss models in linear elastic perfect plasticity its performance is demonstrated for a plane strain benchmark problem

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

transfer operators for evolving meshes in small strain elasto placticity abstract together with error evaluation and mesh refinement transfer operators represent the essential component of the adaptive procedure applied to history dependent materials aspects of the transfer operation discussed in this work relate to equilibrium conditions consistency with the constitutive equation geometric issues and diffusion of state variables between successive meshes in this paper attention is restricted to small strain rate independent elasto plasticity numerical examples are presented to illustrate some practical features of the computational procedure

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

the integral equation method for a steady kinematic dynamo problem with only a few exceptions the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method however the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics the paper is intended to facilitate the use of integral equation solvers in dynamo theory in concrete the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter for the case of spherical geometry three examples of the function r with steady and oscillatory solutions are considered a convergence rate proportional to the inverse squared of the number of grid points is achieved based on this method a convergence accelerating strategy is developed and the convergence rate is improved remarkably typically quite accurate results can be obtained with a few tens of grid points in order to demonstrate its suitability for the treatment of dynamos in other than spherical domains the method is also applied to x2 dynamos in rectangular boxes the magnetic fields and the electric potentials for the first eigenvalues are visualized

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

numerical solutions of the korteweg di vries equation using the periodic scattering transform m representation abstract we develop a numerical procedure based upon the periodic scattering transform for determining the spatial and temporal evolution of solutions to the korteweg de vries kdv equation with periodic boundary conditions in this approach solutions to kdv are found in terms of the representation i e the wave amplitude is given by a trace formula consisting of a linear superposition of hyperelliptic functions j x t 1 j n which are governed by n nonlinearly coupled ordinary differential equations our approach consists of the numerical integration of these equations using a second order taylor series expansion with variable integration step size it is well known that the j are generalizations of elliptic functions to the hyperelliptic class nevertheless we find that the j x t bear little or no resemblance to elliptic functions except in the small amplitude linear limit or for the case of a single soliton we study in particular the most difficult numerical case which occurs for the n soliton wave train solution of the periodic kdv equation

integral equations a tool to solve the schrodinger equation from the integral equation equivalent to the one dimensional schrodinger equation with the boundary conditions appropriate for a bound state problem an exact three point integration rule is derived the approximate evaluation of this rule by means of the euler mclaurin sum rule gives rise to various o h2 and o h4 integration methods associated with the different ways of splitting the starting equation into a free part and an interaction term the method is particularly useful to deal with the singularities of the potential

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

averaging techniques for the effective numerical solution of symm s integral equation of the first kind averaging techniques for finite element error control occasionally called zz estimators for the gradient recovery enjoy a high popularity in engineering because of their striking simplicity and universality one does not even require a pde to apply the nonexpensive post processing routines recently averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method this paper establishes a class of averaging error estimators for boundary integral methods symm s integral equation of the first kind with a nonlocal single layer integral operator serves as a model equation studied both theoretically and numerically we introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order the higher order terms depend on the regularity of the exact solution several numerical experiments illustrate the theoretical results and show that the normally unknown error is sharply estimated by the proposed estimators i e error and estimators almost coincide

coupling of fem and bem for interface problems in viscoplasticity and plasticity with hardening this paper is concerned with three dimensional interface or transmission problems in solid mechanics which consist of the quasi static equilibrium condition in a bounded lipschitz domain omega a first order evolution inclusion in omega and the homogeneous linear elasticity problem in an unbounded exterior domain omega 2 the evolution problem in omega models viscoplasticity and prandtl reus plasticity with hardening in a former paper siam j math anal 25 1994 pp 1468 1487 the author stated the interface problem as well as an equivalent formulation and proved existence and uniqueness of solutions here the numerical approximation of the solutions is considered via a symmetric coupling of the fem finite element method and bem boundary element method in space and the generalized midpoint rule in time besides existence uniqueness and uniform boundedness of the solutions of the discrete problems error estimates proving strong convergence of the numerical procedure are gi

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

hidden grassmann structure in the xxz model for the critical xxz model we consider the space mathcal w alpha of operators which are products of local operators with a disorder operator we introduce two anti commutative families of operators bf b zeta bf c zeta which act on mathcal w alpha these operators are constructed as traces over representations of the q oscillator algebra in close analogy with baxter s q operators we show that the vacuum expectation values of operators in mathcal w alpha can be expressed in terms of an exponential of a quadratic form of bf b zeta bf c zeta

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

large order behaviour of the strong coupling perturbation expansion for anharmonic oscillators a new formula describing the large order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the hamiltonian is suggested a new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested the branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic sextic octic and decadic oscillators

singular points from taylor series a simple and accurate method is developed for calculating singular points from taylor series it consists of finding the least squares deviation of the taylor coefficients from a proposed asymptotic form sequences are obtained that converge quickly to the closest singularity to the origin some simple mathematical examples and physically interesting eigenvalue problems are discussed to illustrate the procedure the branch points of the eigenvalues for the solutions of period 2 of the mathieu equation and those of the stark shifts for rigid symmetric top molecules which were not obtained before are shown

on the numerical integration of the schrodinger equation numerical tests abstract the various numerical methods obtained in a preceding work in the finite difference scheme are compared among themselves and against the exact solutions of some simple exactly soluble potentials the tests strongly favour the limiting method the discretized plane wave method a discussion of the results in the continuous spectrum is also presented

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

large distance asymptotic behavior of the emptiness formation probability of the xxz spin 1 2 heisenberg chain using its multiple integral representation we compute the large distance asymptotic behaviour of the emptiness formation probability of the xxz spin 1 2 heisenberg chain in the massless regime

accurate eigenvalues and eigenfunctions for quantum mechanical anharmonic oscillators the representation of the taylor expansion of the logarithmic derivative of the wavefunction by means of a pade approximant followed by an appropriate quantization condition proves a powerful way of obtaining accurate eigenvalues of the schrodinger equation in this paper we investigate in detail some of the interesting features of this approach termed riccati pade method rpm by means of its application to anharmonic oscillators we analyse the occurrence of many roots in the neighborhoods of the physical eigenvalues in the weak coupling regime and also obtain accurate coefficients of the strong coupling expansion we finally investigate the global and the local accuracy of the rpm eigenfunctions

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

a new efficient method for calculating perturbation energies using functions which are not quadratically integrable a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable is suggested the key to the method is one simple property of the perturbation equations which to our knowledge has not been noticed until know the use of the resolvent in this method is avoided and the energy spectrum of the unperturbed hamiltonian is not needed in the calculation the resulting method is very simple and straightforward and gives very accurate eigenvalues and wavefunctions

numerical analysis of the primal problem of elastoplasticity with hardening the finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time step in an elastoplastic evolution problem in this paper we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of hencky plasticity the finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp piecewise constant trial functions an a priori error estimate is established for the fully discrete method which shows linear convergence as the mesh size tends to zero provided the exact displacement field u is smooth near the boundary of the plastic domain which is unknown a priori it is most likely that u is non smooth in this situation automatic mesh refinement strategies are believed to improve the quality of the finite element approximation we suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate this estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above the constants depend on the hardening involved and become larger for decreasing hardening

integral equation method for the continuous spectrum radial schrodinger equation a new approach to the numerical solution of boundary value problems for differential equations which originated in recent papers by greengard and rokhlin is improved and adapted to the numerical solution of the radial schrodinger equation the approach is based on the conversion of the differential equation into an integral equation together with the application of a spectral type clenshaw curtis quadrature method through numerical examples the integral equation method is shown to be superior to finite difference methods

all first order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable all first order averaging or gradient recovery operators for lowest order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic elliptic model problem in a bounded lipschitz domain in r d given a piecewise constant discrete flux p h e p h that is the gradient of a discrete displacement as an approximation to the unknown exact flux p that is the gradient of the exact displacement recent results verify efficiency and reliability of m min p h q h l 2 q h qh in the sense that m is a lower and upper bound of the flux error p p h l 2 up to multiplicative constants and higher order terms the averaging space q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions the minimal value m is frequently replaced by some averaging operator a p h q h applied within a simple post processing to p h the result q h ap h e q h provides a reliable error bound with m a p h ap h l 2 this paper establishes a c eff m and so equivalence of m and a this implies efficiency of a for a large class of patchwise averaging techniques which includes the zz gradient recovery technique the bound c eff 3 88 established for tetrahedral p 1 finite elements appears striking in that the shape of the elements does not enter the equivalence a m is robust with respect to anisotropic meshes the main arguments in the proof are ascoli s lemma a strengthened cauchy inequality and elementary calculations with mass matrices

averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions averaging techniques or gradient recovery techniques are frequently employed tools for the a posteriori finite element error analysis their very recent mathematical justification for partial differential equations allows unstructured meshes and nonsmooth exact solutions this paper establishes an averaging technique for the hypersingular integral equation on a one dimensional boundary and presents numerical examples that show averaging techniques can be employed for an effective mesh refining algorithm for the discussed test examples the provided estimator estimates the in general unknown error very accurately in the sense that the quotient error estimator stays bounded with a value close to 1