explicit residual based a posteriori error estimation for finite element discretizations of the helmholtz equation computation of the constant and new measures of error estimator quality abstract this paper continues the study of an explicit residual based a posteriori error estimator developed for finite element discretizations of the helmholtz equation in the context of time harmonic exterior acoustics problems in previous papers 1 3 the error estimator was derived and subsequently applied to h adaptive computations of model problems in two dimensions in the present paper a methodology is established for computing error estimates these error estimates are then analyzed in detail the error estimates are made possible by computing the scaling constant which exists in error estimators of this type in our case this constant is difficult to obtain an algorithm for computing the constant is described it is noted that the error indicators used for adaptive computations do not require knowledge of this constant several measures are used to analyze the quality of the error estimates providing a complete description of the error estimator in the context of these problems we compute global effectivity indices and utilize the local element wise effectivity index in computing three additional measures the first of these is the robustness index the others are the modified robustness index and the distribution index which are introduced for the first time it is found that the global effectivity index varies with problem parameters in particular the wave number robustness indices also vary due to the local error definition and wave number values of the distribution index are much better however indicating the suitability of the error estimator in guiding efficient adaptive mesh refinement the analysis of the error estimator is carried out on relatively coarse meshes typical of those used in practical engineering computations

validation of the a posteriori error estimator based on polynomial preserving recovery for linear elements the purpose of this work is to investigate the quality of the a posteriori error estimator based on the polynomial preserving recovery ppr the main tool in this investigation is the computer based theory also a comparison is made between this estimator and the one based on the superconvergence patch recovery spr the results of this comparison were found to be in favour of the estimator based on the ppr copyright 2004 john wiley sons ltd

on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

a posteriori error estimators for the stokes and oseen equations the problem of obtaining posteriori estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed a general approach to the treatment of the constraint condition and to the possible non self adjointness of the associated momentum equations is presented a posteriori error estimates are derived for adaptive h p and h p type finite element schemes key features are that the local error residual problems are not subject to an incompressibility constraint thereby avoiding the need for special finite element schemes and that the analysis is valid for essentially any discretization scheme including continuous and discontinuous pressure spaces the estimator bounds the actual error measured in an energy like norm

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

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

generalization of robustness test procedure for error estimators part i formulation for patches near kinked boundaries in this part of paper we shall extend the formulation proposed by babuska and co workers for robustness patch test for quality assessment of error estimators to more general cases of patch locations especially in three dimensional problems this is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain it has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro patches having found the asymptotic finite element solution the performance of error estimators may be examined although in this paper we focus on the asymptotic behaviour of error estimators the method described in this part may be used to obtain finite element solution for two three dimensional unbounded heat elasticity problems with homogeneous differential equations some numerical results are presented to show the validity and performance of the proposed method copyright 2005 john wiley sons ltd

quaternionic reduction and quaternionic orbifolds on considere un processus de reduction pour des varietes de kahler quaternioniques on montre qu il existe dans ce cas une certaine application impulsion qui se reduit a celle classique quand la courbure scalaire k de l espace est zero et les metriques deviennent hyperkahler quand k 0 cette application impulsion est definie de facon unique pour un groupe compact d automorphismes h

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

η superconvergence of finite element approximations in the interior of general meshes of triangles abstract in this paper we introduce a new definition of superconvergence tne superconvergence which generalizes the classical idea of superconvergence to general meshes we show that this new definition can be employed to determine the regions of least error in any element in the interior of any grid by using a computer based approach we present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat conduction for meshes of conforming triangles of degree p 1 p 5 and elements in the interior of the mesh the results demonstrate that unlike classical superconvergence superconvergence is applicable to the complex grids which are employed in practical engineering computations

superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis abstract in this article we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications however related to adaptive mesh refinement guided by a posteriori error estimators rather few attempts are pursued besides the use of classical residual based error estimators in this article we explore a feature common for isogeometric analysis iga namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the galerkin discretization by utilizing the concept of structured mesh refinement using lr b splines our aim is to facilitate superconvergence behavior for locally refined meshes as well superconvergence behavior matches well with the use of recovery based a posteriori estimator in the superconvergent patch recovery spr procedure however to our knowledge so far the spr procedure has not been exploited in the iga community we start out by addressing the existence of derivative superconvergent points in the computed finite element solution based on b splines and lr b splines for an elliptic model problem 1d and 2d poisson then we present some recovery procedures for improving the derivatives or gradient of the isogeometric finite element solution where the spr procedure will be the main focus in particular we show that our spr procedure for the improvement of derivatives fulfills the desired consistency criteria at the end we develop a posteriori error estimator where the improved gradient obtained from the proposed recovery procedures is used numerical results are presented to illustrate the efficiency of using spr procedure for the improvement of derivatives or gradient of computed solution in isogeometric analysis then the proposed a posteriori error estimator based adaptive refinement methodology is tested to solve smooth and non smooth elliptic benchmark problems the focus is put on whether optimal convergence rates are obtained in the computed solution or not as well as the effectivity index of the proposed error estimators

equilibrated patch recovery error estimates simple and accurate upper bounds of the error this paper introduces a new recovery type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error the upper bound property requires the recovered solution to be both statically equilibrated and continuous the equilibrium is obtained locally patch by patch and the continuity is enforced by a postprocessing based on the partition of the unity concept this postprocess is expected to preserve the features of the locally equilibrated stress field nevertheless the postprocess phase modifies the equilibrium which is no longer exactly fulfilled a new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium this requires computing the 2 norm of the error or relating it with the energy norm the guaranteed upper bounds are obtained by using a pessimistic bound of the error 2 norm derived from an eigenvalue problem nevertheless these bounds are not sharp an additional strategy based on a more accurate assessment of the error 2 norm is introduced providing sharp estimates which are practical upper bounds as it is demonstrated in the numerical tests copyright 2006 john wiley sons ltd

on the asymptotic exactness of bank weiser s estimator in this paper we analyze an error estimator introduced by bank and weiser we prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes by considering a simple example we show that this is not true for general meshes

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

analysis of the zienkiewicz zhu a posteriori error estimator in the finite element method an a posteriori error estimator for finite element analysis proposed by zienkiewicz and zhu is analysed and shown to be effective and convergent in addition we analyse wider classes of estimators of which the zienkiewicz zhu estimator is a special case it is shown that some of these estimators will be asymptotically exact numerical evidence is presented supporting the analysis

a posteriori estimators for nonlinear elliptic partial differential equations many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations for each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible usually the available estimators may be classified into two types i e the implicit estimators based on the solution of a local problem and the explicit estimators based on some suitable norm of the residual in a dual space regarding the performance an estimator is called asymptotically exact if it is a higher order perturbation of a norm of the exact error nowadays one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems the situation is even worse when the nonlinearities involve the highest derivatives occurring in the pde being considered strongly nonlinear pdes in this work we establish conditions under which those estimators originally developed for linear problems may be used for strongly nonlinear problems and how that could be done we also show that under some suitable hypothesis the estimators will be asymptotically exact whenever they are asymptotically exact for linear problems those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems

a posteriori error analysis of finite element solutions for one dimensional problems the paper develops a theory of a posteriors error estimates under the l p energy norm for 2 leqq p leqq infty the theory is based on a general concept of error indicators and error estimators several specific examples of these quantities are introduced and analyzed in detail the results provide a variety of easily computable error estimates under all these important norms for finite element solutions of one dimensional problems

an anisotropic error indicator based on zienkiewicz zhu error estimator application to elliptic and parabolic problems the anisotropic error indicator presented in m picasso comm numer methods engrg 19 2003 pp 13 23 in the frame of the laplace equation is extended to elliptic and parabolic problems our error indicator is derived using the anisotropic interpolation estimates of l formaggia and s perotto numer math 89 2001 pp 641 667 l formaggia and s perotto numer math 2002 doi 10 1007 s002110200415 together with a zienkiewicz zhu error estimator to approach the error gradient a numerical study of the effectivity index is proposed for elliptic diffusion convection and parabolic problems an adaptive algorithm is implemented aimed at controlling the relative estimated error

constitutive relation error estimators for visco plastic finite element analysis with softening a posteriori error estimators based on constitutive relation residuals have been developed for 20 years in particular at cachan this approach has a strong physical meaning and is quite general here we introduce an extended constitutive relation error estimators family able to measure the quality of finite element computations of structures which exhibit plastic viscoplastic behavior with softening these measures take into account over the studied time interval all the classical error sources involved in the computation the space discretization the mesh the time discretization and the iterative technique used to solve the nonlinear discrete problem

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

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

hierarchic models for laminated composites the principles governing the formulation of hierarchic models for laminated composites are discussed the essential features of the hierarchic models described herein are a the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness and b the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity examples are presented

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

basic problems of a posteriori error estimation abstract this paper addresses basic questions regarding the reliability of a posteriori error estimation it analyzes in detail an a posteriori estimator for linear elements on triangular meshes it gives the bounds of the effectivity index depending on the geometry of the triangles and smoothness of the approximated solution the theoretical results are in concordance with the results of our numerical experiments the second part analyzes a one parametric family of estimators in the one dimensional setting and their optimal selection for adaptively constructed meshes

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

a posteriori error estimation for the semidiscrete finite element method of parabolic differential equations in this paper three a posteriori error estimators of the error in the semidiscrete finite element solution discrete in space and continuous in time of parabolic partial differential equations are analyzed this approach is based on a posteriori error estimators for the elliptic pdes it is proven that guaranteed resp asymptotically exact a posteriori error estimator for the elliptic problem yields the guaranteed resp asymptotically exact estimator for the parabolic problem

a posteriori estimation of the error in the error estimate in this paper we address the problem of a posteriori estimation of the error in the error estimate we consider the case of estimates for the error in the derivatives the strains or the stresses which are constructed in terms of locally computed element error indicators of the element residual or the least squares recovery type the estimates of the error in the error estimate have the same structure as the original error estimates and are determined by locally averaging recycling the original error indicators the most accurate indicators of the error in the error indicators are obtained by employing a harmonic basis in the recycling of the indicators namely a basis which locally satisfies the partial differential equation and the boundary conditions

a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification a posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy space discretization with continuous piecewise linear finite elements is considered the l 2 in time h 1 in space error is bounded above and below by an error estimator based on the equation residual numerical results show that the effectivity index is close to one an adaptive finite element algorithm is proposed and a solutal dendrite is computed c 2002 elsevier science b v all rights reserved

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

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

new advances on a posteriori error on constitutive relation in f e analysis this paper deals with error estimators based on residuals on the constitutive relation which have been developed for the past 20 years at cachan especially in the case of elasticity these error estimators can become poor in situations where the material and the mesh are strongly anisotropic the crucial point herein is the construction of equilibrated stress fields from the computed finite element solution therefore in this paper the approach and in particular this last point are re examined several important modifications are introduced in the concepts and basic techniques hence a new generation of error estimators which seem to be extremely robust upon initial applications has been generated

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

asymptotically exact error estimators for rectangular finite elements this paper deals with a posteriors error estimates for finite element approximations of elliptic problems based on superconvergence results a general approach is introduced to construct under reasonable assumptions asymptotically exact error estimators finally several examples are presented and it is shown that other well known estimators can be analyzed in this framework the estimators introduced are of a local character and therefore inexpensive to compute

recursive quadrature mirror filters criteria specification and design method criteria are found for a pair of recursive infinite impluse response iir filters which have mirror image amplitude responses and whose outputs are in phase quadrature at all frequencies a method is suggested for designing filters with these characteristics and their performance and processing requirements are compared with a typical nonrecursive finite impluse response fir realization a particular filter is described in more detail and it is shown how it could be incorporated in a multiband processing scheme it is concluded that recursive quadrature mirror filters could offer considerable savings in terms of signal processing compared with nonrecursive filters with similar performances additionally it is suggested that because they can be designed to have relatively short absolute delays the performance of the iir filters may prove superior for certain applications

the problem of the selection of an a posteriori error indicator based on smoothening techniques abstract this paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution an error estimate based on local l projections is analyzed in the case of translation invariant meshes it is shown that for general meshes this technique does not lead to an asymptotically exact estimator the problem is analyzed in detail in the one dimensional setting it is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth an optimal estimator for adaptively constructed meshes is given finally a general mathematical framework for the quality assessment of estimators is introduced

a posteriori error estimation based on the superconvergent recovery by compatibility in patches the present work deals with an a posteriori error estimator for linear finite element analysis based on a stress recovery procedure called recovery by compatibility in patches the key idea of this procedure is to recover improved stresses by minimizing the complementary energy over patches of elements displacements computed by the finite element analysis are prescribed on the boundary of the patch here a new form of this recovery procedure is presented adopting a different patch configuration centred upon an element instead of a node allows to drastically simplify the recovery process thus improving efficiency and making the implementation in finite element codes much easier the robustness tests demonstrate that the error estimator associated to the new form of the recovery procedure retains the very good properties of the original one such as superconvergence the numerical results on two common benchmark problems confirm the effectiveness of the proposed error estimator which appears to be competitive with those currently available copyright 2006 john wiley sons ltd

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

on one approach to a posteriori error estimates for evolution problems solved by the method of lines in this paper we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines one of our goals is to apply known estimates derived for elliptic problems to evolution equations we apply the new technique to three distinct problems a general nonlinear parabolic problem with a strongly monotonic elliptic operator a linear nonstationary convection diffusion problem and a linear second order hyperbolic problem the error is measured with the aid of the l 2 norm in the space time cylinder combined with a special time weighted energy norm theory as well as computational results are presented

constitutive relation errors for f e analysis considering visco plasticity and damage a posteriori error estimators based on constitutive relation residuals are developed to measure the quality of finite element computations of structures which exhibit plastic viscoplastic behaviour with softening all the classical error sources involved in the computation over the studied time interval are taken into account the space discretization the mesh the time discretization and the iterative technique used to solve the non linear discrete problem copyright 2001 john wiley sons ltd

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

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

on an h type mesh refinement strategy based on minimization of interpolation errors abstract an adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre assigned limit the estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives applications to several two dimensional model problems are discussed the results indicate that the method can be very effective for both regular problems and problems with strong singularities

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

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

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

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

accurate recovery based upper error bounds for the extended finite element framework this paper introduces a recovery type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method xfem the paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a fem framework the upper bound property requires the recovered solution to be equilibrated and continuous the proposed technique consists of using a recovery technique especially adapted to the xfem framework that yields equilibrium at a local level patch by patch then a postprocess based on the partition of unity concept is used to obtain continuity the result is a very accurate but only nearly statically admissible recovered stress field with small equilibrium defaults introduced by the postprocess sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults as demonstrated by the numerical tests

generalization of robustness test procedure for error estimators part ii test results for error estimators using spr and rep in this part of the paper we shall use the formulation given in the first part to assess the quality of recovery based error estimators using two recovery methods i e superconvergent patch recovery spr and recovery by equilibrium in patches rep the recovery methods have been shown to be asymptotically robust and superconvergent when applied to two dimensional problems in this study we shall examine the behaviour of the recovery methods on several three dimensional mesh patterns for patches located either inside or at boundaries this is performed by first finding an asymptotic finite element solution irrespective of boundary conditions at far ends of the domain and then applying the recovery methods the test procedure near kinked boundaries is explained in a step by step manner the results are given in a series of tables and figures for various cases of three dimensional mesh patterns it has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

a posteriori estimation of the error in the recovered derivatives of the finite element solution this work addresses the accuracy of the solution derivatives which are recovered by local averaging of the finite element solution the main results of the study are 1 the error in the locally averaged derivatives e g the derivatives which are recovered by the zienkiewicz zhu superconvergent patch recovery zz spr or other similar local recoveries can be more than the error in the derivatives computed directly from the finite element solution especially in the case of unsmooth solutions and or coarse meshes 2 in order to determine which solution derivatives should be relied upon the locally averaged ones or the ones computed directly from the finite element solution one must be able to estimate their errors it is shown that one can obtain indicators of the error in the derivatives recovered by the zz spr by employing an additional local averaging of the recovered derivatives recycling of the zz spr or by comparing the derivatives computed by the zz spr with the derivatives obtained using a different local averaging which takes into account the character of the exact solution harmonic averaging

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

the finite element patch test revisited a computer test for convergence validation and error estimates abstract the patch test was introduced in the mid 1960s first as a check of satisfactory performance for elements violating continuity requirements and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors in its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct later it was established that the test could also be used to find the convergence order of any finite element type while consistency is one of the necessary requirements for convergence the second is of course that of stability in the original early work only displacement type structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms in a 1986 reformulation of the patch test this stability or solvability requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms 1 2 although some misinterpretation of the patch test has occurred in literature 3 4 the authors believe that with certain restrictions on regularity of the problem dealt with the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical babuska brezzi criteria 5 6 in recent years another important use of the patch test has been found by babuska and his coworkers 7 9 this work tests the asymptotic effectivity of various error estimating procedures used with various element patterns

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

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

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