prolongation structures of complex quasi polynomial evolution equations the authors use wahlquist estabrook prolongation theory 1976 to investigate second order complex equations of generalised nls dnls type they isolate a number of new integrable cases

some improvements to the flux type a posteriori error estimators abstract this paper presents explicit equilibrated fluxes for the local pure neumann problems used by the a posteriori error estimator in ainsworth and oden m ainsworth j t oden a procedure for a posteriori error estimation for h p finite element methods comput methods appl mech engrg 101 1992 73 96 we modify the target function for minimization to obtain new equilibrated fluxes in a weighted form we investigate the performance of the new equilibrated fluxes in two dimensional spectral hp element approximations of an elliptic equation using conforming triangle elements the new fluxes appear more robust and lead to substantial computational savings compared to the original numerical procedure in ainsworth and oden ainsworth and oden et al 1992

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

local and pollution error estimation for finite element approximations of elliptic boundary value problems this paper addresses the issue of local elementwise error estimation of finite element approximations of elliptic boundary value problems the characterization of element error as local and pollution error component is presented and the relationship between energy norms of local errors and that predicted by means of a posteriori error estimators is investigated in addition techniques for calculating element indicators of local error pollution error and other error indicators are presented

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

patch recovery based on superconvergent derivatives and equilibrium in this paper a postprocessing technique is developed for determining first order derivatives fluxes stresses at nodal points based on derivatives in superconvergent points it is an extension of the superconvergent patch recovery technique presented by zienkiewicz and zhu in contrast to that technique all flux or stress components are interpolated at the same time coupled by equilibrium equations at the superconvergent points the equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

factorisation of backlund transformations the factorisation of the l operator in a lax pair l p has played an important role in the theory of integrable systems for example the miura transformation can be immediately obtained from the factorisation of the schrodinger operator delta x2 u k2 delta x a delta x b requires that u k2 ax a2 and a b for zero curvature representations this procedure is not available however the author attempts to produce an analogous theory of factorisations for zero curvature representations

a refined global local finite element analysis method a refined global local method was proposed to improve the efficiency of finite element analysis the proposed method was based on the regular finite element method in conjunction with three basic step i e the global analysis the local analysis and the refined global analysis in the first two steps a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest these local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses in the third step a new global displacement distribution based on the results of the previous two steps was assumed for the analysis from which much improved solutions for both stresses and displacements were produced numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method further this method is suitable for parallel computation

the prolongation structures of quasi polynomial flows we look closely at the process of finding a wahlquist estabrook prolongation structure for a given system of nonlinear evolution equation s there are two main steps in this calculation the first to reduce the problem to the investigation of a finitely generated free lie algebra with constraints the second to find a finite dimensional linear representation of these generators we discuss some of the difficulties that arise in this calculation for quasi polynomial flows defined later we give an algorithm for the first step we do not totally solve the problems of the second step but do give an algebraic framework and a number of techniques that are quite generally applicable we illustrate these methods with many examples several of which are new

perfect cocycles through stochastic differential equations we prove that if is a random dynamical system cocycle for whicht t x is a semimartingale then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment helix and conversely this relation is succinctly expressed as semimartingale cocycle exp semimartingale helix to implement it we lift stochastic calculus from the traditional one sided time to two sided timet and make this consistent with ergodic theory we also prove a general theorem on the perfection of a crude cocycle thus solving a problem which was open for more than ten years

an a posteriori error estimate for finite element approximations of a singularly perturbed advection diffusion problem abstract in this paper the author presents an a posteriori error estimator for approximations of the solution to an advection diffusion equation with a non constant vector valued diffusion coefficient in a conforming finite element space based on the complementary variational principle we show that the error of an approximate solution in an associated energy norm is bounded by the sum of the weighted l 2 norms of solutions to a set of independent complementary variational problems each defined on only one element of the partition this error bound guarantees the over estimation of the true error and does not depend unfavourably on as goes to zero although the original equation is a non self adjoint problem the strong form of each local variational problem is always a poisson equation with neumann boundary conditions the approximation of these local problems is then discussed and it is shown that omitting a higher order term the finite element solutions of these local complementary variational problems provide a computable upper error bound for the original finite element approximation in the energy norm numerical results presented to validate the theoretical results show that the computed error bounds are tight for a wide range of values of and always over estimate the true errors

integrability of kersten krasil shchik coupled kdv mkdv equations singularity analysis and lax pair the integrability of a coupled kdv mkdv system is tested by means of singularity analysis the true lax pair associated with this system is obtained by the use of prolongation technique

a simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems abstract a simple flux reconstruction for finite element solutions of reaction diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction diffusion problem the flux reconstruction is based on simple independent post processing operations over patches of elements in conjunction with standard raviart thomas vector fields we prove that the corresponding local error indicators are locally efficient and robust up to a nonstandard oscillation term with respect to any mesh size and any size of the reaction coefficient including the singularly perturbed limit

integrability of a coupled kdv system painleve property lax pair and backlund transformation the integrability of a coupled kdv system is studied by prolongation technique and singularity analysis as a result backlund transformation and linear spectral problem associated with this system are obtained some special solutions of the system are also proposed

robust error bounds for finite element approximation of reaction diffusion problems with non constant reaction coefficient in arbitrary space dimension abstract we present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension the estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient including the singularly perturbed case the approach is based on robustly equilibrated boundary flux functions of ainsworth and oden 2000 and on subsequent robust and explicit flux reconstruction this paper simplifies and extends the applicability of the previous result of ainsworth and vejchodsk 2011 in three aspects i arbitrary dimension ii mixed boundary conditions and iii non constant reaction coefficient it is the first robust upper bound on the error with these properties an auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices

a posteriori estimates of the solution error caused by discretization in the finite element finite difference and boundary element methods a theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems this method gives bounded global estimates of the error in the energy norm pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence the versatility of this method is illustrated by its application to numerical solutions from finite element finite difference and boundary element methods

self adaptive stress analysis based on stress convergence a new algorithm for self adaptive stress analysis is presented the algorithm is based on h convergence and uses the measured stress convergence rate as an error indicator a theoretical relation between stress convergence rate and pointwise error of stresses is derived for both the 1 d and 2 d case some examples are presented to illustrate the behaviour of the algorithm

stability of a stochastic two dimensional non hamiltonian system we study the top lyapunov exponent of the response of a two dimensional non hamiltonian system driven by additive white noise the origin is not a fixed point for the system however there is an invariant measure for the one point motion of the system in this paper we consider the stability of the two point motion by khasminskii s method of linearization along trajectories the specific system we consider is the third order truncated normal form of the unfolding of a hopf bifurcation we show that in the small noise limit the top lyapunov exponent always approaches zero from below and is thus negative for noise sufficiently small we also show that there exist open sets of parameters for which the top lyapunov exponent is positive thus the two point motion can be either stable or unstable while the stationary density that describes the one point motion always exists

prolongation structures and exact solutions of k m n equations in this paper rosenau and hyman s phys rev lett 70 564 1993 k m n equations are studied by prolongation technique it is proved that k m n equations are lax integrable only for certain special parameters m n the nontrivial prolongation structures and lax pairs for the integrable cases are given finally as an example the one and two soliton solutions for the k 12 12 equation are derived by means of inverse scattering method

miura type transformations and homogeneous spaces we relate miura type transformations mts over an evolution system to its zero curvature representations with values in lie algebras we prove that certain homogeneous spaces of produce mts and show how to distinguish these spaces for a scalar translation invariant evolution equation this allows us to classify all mts in terms of homogeneous spaces of the wahlquist estabrook algebra of the equation for other evolution systems this allows us to construct some mts as an example we study mts over the kdv equation a fifth order equation of harry dym type and the coupled kdv mkdv system of kersten and krasilshchik

element by element a posteriori error estimation and improvement of stress solutions for two dimensional elastic problems a quantitative method of a posteriori error estimation based on finite element solutions is developed the proposed method can estimate the error of solutions obtained by the finite element method for two dimensional elastic problems with 4 node elements since the error is estimated element by element the proposed method does not require a large memory or long computing time not only rectangular elements but also arbitrarily shaped 4 node elements can be used in this method for estimating the error in the finite element computation with high accuracy the finite element solutions are improved by adding the estimated errors onto the original solutions the proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used

the background of error estimation and adaptivity in finite element computations the paper summarizes the present state of the art concerning error estimation and adaptive remeshing recent work shows that methods based on recovery result on most accurate error estimators

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

integrability of the coupled kdv equations derived from two layer fluids prolongation structures and miura transformations abstract the lax integrability of the coupled kdv equations derived from two layer fluids s y lou b tong h c hu x y tang coupled kdv equations derived from two layer fluids j phys a math gen 39 2006 513 527 is investigated by means of prolongation technique as a result the lax pairs of some painleve integrable coupled kdv equations and several new coupled kdv equations are obtained finally the miura transformations and some coupled modified kdv equations associated with the lax integrable coupled kdv equations are derived by an easy way

asymptotic solutions for predicted natural frequencies of two dimensional elastic solid vibration problems in finite element analysis in order to assess the discretization error of a finite element solution asymptotic solutions for predicted natural frequencies of two dimensional elastic solid vibration problems in the finite element analysis are presented in this paper since the asymptotic solution is more accurate than the original finite element solution it can be viewed as an alternative solution against which the original finite element solution can be compared consequently the discretization error of the finite element solution can be evaluated due to the existence of two kinds of two dimensional problems in engineering practice both the plane stress problem and the plane strain problem have been considered and the corresponding asymptotic formulae for predicted natural frequencies of two dimensional solids by the finite element method have been derived from the fact that a discretized finite element system approaches a continuous one if the finite element size approaches zero it has been demonstrated from the related numerical results of three examples that the present asymptotic solution which can be obtained by simply using the corresponding formula without any further finite element calculation is indeed more accurate than the original finite element solution so that it can be considered as a kind of corrected solution for the discretization error estimation of a finite element solution

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

maximal lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise asymptotic expansions for the exponential growth rate known as the lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise are constructed such systems arise naturally in the investigation of the stability of steady state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems almost sure stability or instability of dynamical systems depends on the sign of the maximal lyapunov exponent stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable the results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging where approximate ito equations in amplitudes and phases are obtained in the sense of weak convergence

scalar second order evolution equations possessing an irreducible sl2 valued zero curvature representation we find all scalar second order evolution equations that possess an sl2 valued zero curvature representation irreducible to a proper subalgebra of sl2 none of these zero curvature representations depends on a parameter that could serve as the spectral parameter

a subdomain residual method for a posteriori error estimates of finite element solutions of viscoelastic fluid flows abstract in this paper we present a subdomain residual method which gives a posteriori estimates of the errors in finite element solutions of viscoelastic fluid flows we employ a coupled approach in which the estimated errors in both velocity and extra stress are computed simultaneously by solving a series of local problems the subdomains on which the local problems are posed consist of patches of elements a given parameter which is independent of the element size is employed to determine the size of the subdomains to ensure that the error estimate improves as the element size is reduced the error estimator is applied to finite element solutions of newtonian and non newtonian fluid flows past a cylindrical tube in a duct the contour plots of the components of the estimated errors in velocity and extra stress are presented these errors are compared with reference errors

a tutorial in elementary finite element error analysis a systematic presentation of a priori and a posteriori error estimates abstract this paper provides a tutorial on the derivation of a priori and explicit residual based a posteriori error estimates for galerkin finite element discretizations of general linear elliptic operators for simplicity the presentation is in one dimension although extensions to multidimensions should be straightforward the structures of explicit residual based a posteriori error estimators and a priori error estimators are very similar they are each in the form of an upper bound on the error measured in some norm the structures of their derivations also follow similar lines it is therefore appropriate to present and discuss a priori and a posteriori error estimators in parallel in this paper error bounds are obtained for the error measured in the energy norm and the l2 norm in addition symmetric nonsymmetric positive definite and indefinite operators are discussed a priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm special attention is given to the advection diffusion equation a nonsymmetric operator and the helmholtz equation an indefinite operator we systematically describe the necessary steps in deriving a priori and a posteriori error estimates and provide a simplified understanding of their differences as well as their similarities

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

stochastic bifurcations in a bistable duffing van der pol oscillator with colored noise this paper aims to investigate gaussian colored noise induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable duffing van der pol oscillator by using the stochastic averaging method theoretically one can obtain the stationary probability density function of amplitude for the duffing van der pol oscillator and can reveal interesting dynamics under the influence of gaussian colored noise stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution which indicates that system parameters noise intensity and noise correlation time respectively can be treated as bifurcation parameters they also imply that the effects of multiplicative noise are rather different from that of additive noise the results of direct numerical simulation verify the effectiveness of the theoretical analysis moreover the largest lyapunov exponent calculations indicate that p and d bifurcations have no direct connection

a posteriori error estimators for second order elliptic systems part 2 an optimal order process for calculating self equilibrating fluxes abstract we present full details of an algorithm which can be used to obtain self equilibrating fluxes for arbitrary h p finite element approximations all of the calculations are performed on a local scale yielding an optimal order process o n where n is the number of elements in the mesh owing to the local nature of the process the algorithm is perfectly suited to parallel implementation the algorithm can be applied in both two and three dimensions for unstructured k irregular meshes and elements of non uniform degree finally it is shown that the implementation of the flux equilibration algorithm can take advantage of calculations already performed in obtaining the finite element approximation itself

closed form stiffness matrices and error estimators for plane hierarchic triangular elements closed form expressions for the stiffness matrix and a simple error estimator and error indicator are derived for plane straight sided triangular finite elements in elasticity problems the calculation of the error estimator is performed on an element by element basis and is found to be very accurate and efficient in general the solutions for benchmark problems using the error indicators for selective refinement of the regions show accelerated convergence when compared to the convergence rate of solutions using uniform mesh refinement evaluation of the stiffness matrices and error estimators using explicit formulations is found to be several times faster than numerical integration

a posteriori error estimators for second order elliptic systems part 1 theoretical foundations and a posteriori error analysis abstract this is the first in a series of two papers dealing with a posteriori error estimation for h p finite element approximation of second order elliptic systems in this paper we shall present the fundamental ideas behind the error estimator and the theoretical foundations for the method the second paper deals with the algorithmical details

the asymptotic stability of a noisy non linear oscillator abstract the purpose of this work is to obtain an approximation for the top lyapunov exponent the exponential growth rate of the response of a single well kramers oscillator driven by either a multiplicative or an additive white noise process to this end we consider the equations of motion as dissipative and noisy perturbations of a two dimensional hamiltonian system a perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation we show analytically that the top lyapunov exponent is positive and for small values of noise intensity e and dissipation e the exponent grows in proportion with e1 3

korteweg de vries burgers equation and the painleve property it is shown that the korteweg de vries burgers equation possesses the painleve property conditionally using an algorithmic approach a travelling wave solution is reproduced

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

computing bounds for linear functionals of exact weak solutions to poisson s equation we present a method for poisson s equation that computes guaranteed upper and lower bounds for the values of piecewise polynomial linear functional outputs of the exact weak solution of the infinite dimensional continuum problem with piecewise polynomial forcing the method results from exploiting the lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem localization is achieved by lagrangian relaxation and the bounds are computed by appeal to a local dual problem the proposed method computes approximate lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers the computed bounds hold uniformly for any level of refinement and in the asymptotic convergence regime of the finite element method the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution given a finite element solution and its output adjoint solution the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization the elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators as we demonstrate with a numerical example

a procedure for a posteriori error estimation for h p finite element methods abstract a new approach to a posteriori error estimation is outlined which is applicable to general h p finite element approximations of general classes of boundary value problems the approach makes use of duality arguments and is based on the element residual method erm important aspects of the method are that it provides a systematic approach toward deriving element boundary conditions for the erm it leads to an upper bound for the global error in an appropriate energy norm and it is valid for non uniform and irregular h p meshes in the present exposition a brief outline of the theoretical foundations of the method is given together with the results of its application to several representative problems these results show that the approach is applicable to general linearly elliptic systems including unsymmetrical operators and that the method is valid for broad classes of linear and non linear problems

the equilibrium method for local a posteriori pointwise estimation of the finite element error the equilibrium method for the recovery of finite element errors is extended and given the theoretical foundation necessary to employ it to the pointwise recovery of the finite element function the practical implications of the resulting theory is studied extensively through a computational example

error estimators using global local methods error estimators were derived using global local and refined global local finite element methods and an extrapolation technique the proposed error estimators are to be used in estimating the global error energy norm local error energy norm and nodal displacement error these error estimators were applied to three numerical examples the results clearly demonstrated that these error estimators are of high quality

a posteriori pointwise upper bound error estimates in the finite element method an error estimate for the finite element method is presented in this paper the error is identified as the response to a set of residual forces and a complementary analysis provides an upper bound estimate of the global energy of the error the inequality proposed by babuska and miller1 is then employed to bound the error in stress and displacement at a point the formula is derived for two dimensional elasticity but the procedure is general and can be applied to three dimensional and other problems numerical experiments using the procedure are carried out and the results are given for the four node bilinear compatible element and plane stress

the computation of bounds for linear functional outputs of weak solutions to the two dimensional elasticity equations we present a method for the computation of upper and lower bounds for linear functional outputs of the exact solutions to the two dimensional elasticity equations the method can be regarded as a generalization of the well known complementary energy principle the desired output is cast as the supremum of a quadratic linear convex functional over an infinite dimensional domain using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary or dual problem in order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced first the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle the lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation second the solution of the adjoint problem is approximated over the triangulation using a standard galerkin finite element approach a lower bound for the output of interest is computed by repeating the process for the negative of the output reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output the method can be easily generalized to three dimensions however a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions the computed bound gaps are found to converge optimally that is at the same rate as the finite element approximation an attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds using this data set and a simple algorithm the correctness of the computed bounds can be established without recourse to the original code used to compute them in the present paper only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered two examples are given to illustrate the proposed methodology

the relationship of some a posteriori estimators we present analytical and numerical investigation in the relationship of the recovery error estimator and the implicit residual error estimator it is shown that analytically both error estimators are equivalent for one dimensional problems numerical study indicates that such equivalence also exist for two dimensional problems

pointwise and local error estimates for the quantities of interest in two dimensional elasticity abstract in this paper a technique is presented to obtain pointwise and local a posteriori error estimates for the quantities of interest in finite element approximations to linear elasticity problems two numerical experiments on stress concentration problems are presented to validate the analysis they show that useful error estimates on the global energy of the error and on the local peak stresses can be achieved with coarse meshes

lyapunov exponents for small random perturbations of hamiltonian systems consider the stochastic nonlinear oscillator equation ddot x x x 3 varepsilon 2 beta dot x varepsilon sigma x dot w t with beta 0 and sigma neq 0 if 4 beta sigma 2 0 then for small enough varepsilon 0 the system x dot x is positive recurrent in bf r 2 setminus 0 0 now let overline lambda varepsilon denote the top lyapunov exponent for the linearization of this equation along trajectories the main result asserts that overline lambda varepsilon varepsilon 2 3 overline lambda o varepsilon 4 3 qquad mbox as varepsilon to 0 with overline lambda 0 this result depends crucially on the fact that the system above is a small perturbation of a hamiltonian system the method of proof can be applied to a more general class of small perturbations of two dimensional hamiltonian systems the techniques used include i an extension of results of pinsky and wihstutz for perturbations of nilpotent linear systems and ii a stochastic averaging argument involving

positive lyapunov exponents in the kramers oscillator the maximum lyapunov exponent is computed numerically for the double well oscillator in a heat bath positive exponents are found in a wide range of friction coefficients in the low damping regime

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

a posteriori error estimation the relationship between different procedures to address the issue of accurate and efficient estimation of the discretization error of the finite element solution the relationship between different types of error estimators is investigated we found that some of the widely used error estimators are in fact related although they were originally derived from different methodologies the recognition of the relation between these error estimators provides a different viewpoint on the theory of a posteriori error estimation and a guidance in the development of the error estimators

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators

on using degenerated solid shell elements in adaptive refinement analysis three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems the stress recovery procedure expressed in a convective patch co ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation the performance of the stress recovery procedure the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems it is found that when adaptive refinement is used the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates it is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations in addition if complete lagrangian polynomial terms are used the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed copyright 1999 john wiley sons ltd

the self equilibration of residuals and complementary a posteriori error estimates in the finite element method it is demonstrated that the residual in a compatible displacement finite element solution can be partitioned into local self equilibrating systems on each element an a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved

a posteriori error estimates and adaptivity for finite element solutions in finite elasticity methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems in contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context in the paper a brief outline of continuum theory of finite elasticity is first given using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally the proposed error estimator is physically interpreted in the energy sense we then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences

stationary energy probability density of oscillators driven by a random external force we derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive gaussian noise our study focuses on two cases i a harmonic oscillator subjected to gaussian colored noise with an arbitrary correlation function and ii nonlinear oscillators with a general potential driven by gaussian white noise we also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy to illustrate our general results we consider specifically the case of exponentially correlated noise for i and power law and bistable potentials for ii our theoretical results are substantiated by langevin simulations between the kinetic energy and the potential energy of the oscillator our studies focus on two cases first we consider a harmonically bound particle driven by general additive gaussian colored noise second we consider a particle in a general potential driven by gaussian white noise in both cases we develop a general theory and study specific illustrative examples the paper is organized as follows in sec ii we formulate the langevin equations for a particle with inertia in a general potential driven by an additive random force we also derive the general relation between the stationary energy pdf and the stationary pdf for the particle s position and velocity section iii is devoted to the motion of a particle in a harmonic potential driven by colored gaussian noise with an arbitrary correlation function the stationary energy pdf as well as the first and second moments of the energy is derived analytically we apply these results to the case of exponentially correlated gaussian noise and to gaussian white noise for these specific cases we also determine the partition of the energy between kinetic and potential energy we analyze the effect of the potential shape on the stationary energy pdf for oscillators driven by an additive gaussian white noise in sec iv w e apply our analytical results to two specific cases a general power law potential and a bistable potential the first type of oscillator without friction and with additive noise 36 and multiplicative noise 37 has recently been studied in the context of anomalous diffusion the second type the duffing oscillator has been investigated as an alternative to the linear oscillator for harvesting energy from external random sources 28 29 35 38 39

the extended estabrook wahlquist method abstract variable coefficient korteweg de vries vckdv modified korteweg de vries vcmkdv and nonlinear schrodinger nls equations have a long history dating from their derivation in various applications a technique based on extended lax pairs has been devised recently to derive time and space dependent coefficient generalizations of various such lax integrable nlpde hierarchies which are thus more general than almost all cases considered earlier via methods such as the painleve test bell polynomials and similarity methods however this technique although operationally effective has the significant disadvantage that for any integrable system with spatiotemporally varying coefficients one must guess a generalization of the structure of the known lax pair for the corresponding system with constant coefficients motivated by the somewhat arbitrary nature of the above procedure we embark in this paper on an attempt to systematize the derivation of lax integrable systems with variable coefficients we consider the estabrook wahlquist ew prolongation technique a relatively self consistent procedure requiring little prior information however this immediately requires that the technique be significantly generalized in several ways including solving matrix partial differential equations instead of algebraic ones as the structure of the lax pair is systematically computed and also in solving the constraint equations to deduce the explicit forms for various coefficient matrices the new and extended ew technique which results is illustrated by algorithmically deriving generalized lax integrable versions of the nls generalized fifth order kdv mkdv and derivative nonlinear schrodinger dnls equations we also show how this method correctly excludes the existence of a nontrivial lax pair for a nonintegrable nlpde such as the variable coefficient cubic quintic nls

a posteriori error estimation of steady state finite element solutions of the navier stokes equations by a subdomain residual method abstract we present a subdomain residual method as well as its mathematical basis for estimating errors in steady state finite element solutions of the incompressible navier stokes equations the estimated errors are obtained by solving a series of local problems in which velocity boundary condition is used wherever the exact traction boundary condition is not available an iterative procedure similar to the newton method is employed to improve the error estimates the performance of the method is demonstrated in two numerical examples i e the channel flow over a backward facing step and past a circular cylinder at low reynolds numbers

prolongation structures and lie algebra real forms a technique useful as an aid for finding finite dimensional representations of a wahlquist estabrook prolongation algebra is introduced the technique is illustrated with an examination of equations of the form 2za x1 x2 fa zb

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

a posteriori error estimates in finite difference techniques abstract a technique is described which defines an upper bound for the effect of the truncation error on finite difference solutions to elliptic boundary value problems this technique does not attempt to provide an estimate which can be used to correct the finite difference solution but rather to produce a guaranteed upper bound measure of the error which can be used to assess the quality of the solution application to two problems is given to demonstrate the basic features of the analysis which is implemented in a post processing mode once the original solution has been completed

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985