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

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors conforming approximations this paper derives a posteriori error estimates for conforming numerical approximations of the laplace eigenvalue problem with a homogeneous dirichlet boundary condition in particular upper and lower bounds for an arbitrary simple eigenvalue are given these bounds are guaranteed fully computable and converge with optimal speed to the given exact eigenvalue they are valid without restrictions on the computational mesh or on the approximate eigenvector we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones which can be checked in practice guaranteed fully computable optimally convergent and polynomial degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well under the same hypotheses remarkably there appears no unknown solution regularity or polynomial degree dependent constant in our theory and no convexity regularity assumption on the computational domain exact eigenvector s is needed two improvements of the multiplicative constant appearing in our estimates are presented first it is reduced by a fixed factor under an explicit a posteriori calculable condition on the mesh and on the approximate eigenvector eigenvalue pair second when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants the multiplicative factor can be brought to the optimal value of one inexact algebraic solvers are taken into account the estimates are valid on each iteration and can serve for the design of adaptive stopping criteria the application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided along with a numerical illustration on a set of test problems

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

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

transmission of information in active networks shannon s capacity theorem is the main concept behind the theory of communication it says that if the amount of information contained in a signal is smaller than the channel capacity of a physical media of communication it can be transmitted with arbitrarily small probability of error this theorem is usually applicable to ideal channels of communication in which the information to be transmitted does not alter the passive characteristics of the channel that basically tries to reproduce the source of information for an active channel a network formed by elements that are dynamical systems such as neurons chaotic or periodic oscillators it is unclear if such theorem is applicable once an active channel can adapt to the input of a signal altering its capacity to shed light into this matter we show among other results how to calculate the information capacity of an active channel of communication then we show that the channel capacity depends on whether the active channel is self excitable or not and that contrary to a current belief desynchronization can provide an environment in which large amounts of information can be transmitted in a channel that is self excitable an interesting case of a self excitable active channel is a network of electrically connected hindmarsh rose chaotic neurons

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

a posteriori error estimators for nonconforming approximation of some quasi newtonian flows this paper deals with a posteriori error estimators for nonconforming approximations of quasi newtonian flows we consider the crouzeix raviart piecewise linear approximations of scalar elliptic problems and define an error estimator when omega is a simply connected domain the error is dominated by the estimator this estimator can be generalized to higher order elements we define a posteriori error estimators for the fortin soulie piecewise quadratic approximations of quasi newtonian flows and prove that the error is dominated by the estimator this estimator can be computed locally in terms of the approximate solution and is therefore suitable for adaptive refinement

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a posteriori error estimates in finite element acoustic analysis we present an a posteriori error estimator for the approximations of the acoustic vibration modes obtained by a finite element method which does not present spurious or circulation modes for non zero frequencies we prove that the proposed estimator is equivalent to the error in the approximation of the eigenvectors up to higher order terms with constants independent of the eigenvalues numerical results for some test examples are presented which show the good behavior of the estimator when it is used as local error indicator for adaptive refinement

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

efficient and reliable hp fem estimates for quadratic eigenvalue problems and photonic crystal applications we present a posteriori analysis of higher order finite element approximations hp fem for quadratic fredholm valued operator functions residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem for the reasons of robustness we also consider approximations of the associated invariant pairs we show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator as a model problem we consider spectral problems arising in analysis of photonic crystals in particular we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple numerical experiments with hp fem show the predicted convergence rates the measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem we also present a benchmark estimator based on the dual weighted residual dwr approach which is more expensive to compute but whose measured effectivities are close to one

on an adaptive stabilized mixed finite element method for the oseen problem with mixed boundary conditions abstract we consider the oseen problem with nonhomogeneous dirichlet boundary conditions on a part of the boundary and a neumann type boundary condition on the remaining part suitable least squares terms that arise from the constitutive law the momentum equation and the dirichlet boundary condition are added to a dual mixed formulation based on the pseudostress velocity variables we prove that the new augmented variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas elements and the velocity is approximated by continuous piecewise polynomials we develop an a posteriori error analysis based on a helmholtz type decomposition and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the dirichlet boundary where they both have two additional terms we prove that these a posteriori error indicators are reliable and locally efficient finally we provide several numerical experiments that support the theoretical results

improved accuracy by adapted mesh refinements in the finite element method for appropriately adapted mesh refinements optimal order error estimates are proved for the finite element approximate solution of the neumann problem for the second order elliptic equation lu 8 where 8 is the dirac distribution

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

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

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

an h adaptive mortar finite element method for finite deformation contact with higher order p extension in this work we present a generalization of the mortar segment to segment method for finite deformations contact to an h adaptive version with possible p extension i e using higher order approximation we recall the main ideas of the mortar algorithm and present the key aspects of adaptivity error estimation and an h adaptive strategy the p extension exploits the feature of the hp adaptive code in which the contact solver is implemented to handle meshes with nodes of nonuniform orders we use it to set interior nodes to higher order while leaving linear boundary contact nodes which can be processed by the standard mortar algorithm accuracy of elements with low order nodes is restored by adequate subdividing of these elements adaptivity and p extension are illustrated with a few numerical tests

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

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

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

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

an anisotropic error estimator for the crank nicolson method application to a parabolic problem in this paper we derive two a posteriori upper bounds for the heat equation a continuous piecewise linear finite element discretization in space and the crank nicolson method for the time discretization are used the error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure the error due to the time discretization is obtained using two different continuous piecewise quadratic time reconstructions the first reconstruction is developed following g akrivis c makridakis and r h nochetto math comp 75 2006 pp 511 531 while the second one is new moreover in the case of isotropic meshes only upper and lower bounds are provided as in r verfurth calcolo 40 2003 pp 195 212 an adaptive algorithm is developed numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one in particular the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

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

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

a posteriori error estimates for some model boundary control problems in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal boundary control problems we derive a posteriori error estimates for both the state and the control approximation on polygonal domains such estimates which are apparently not available in the literature can be used to construct reliable adaptive finite element approximation schemes for the control problems

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

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

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

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