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

recovery type superconvergence and a posteriori error estimates for control problems governed by stokes equations in this paper we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by stokes equations we obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods which are the patch recovery technique and the least squares surface fitting method our results are based on some regularity assumption for the stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions

local and pointwise error estimates of the local discontinuous galerkin method applied to the stokes problem we prove local and pointwise error estimates for the local discontinuous galerkin method applied to the stokes problem in two and three dimensions by using techniques originally developed by a schatz math comp 67 1998 877 899 to prove pointwise estimates for the laplace equation we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries

pointwise error estimates for differences in piecewise linear finite element approximations we consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem new estimates for the differences e x1 e x2 where e x u x u h x is the error and x1 and x2 are any two points in the domain are obtained in terms of weighted l infty norms as a consequence so called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element

on pole assignability over polynomial rings by an explicit counterexample it is shown that the polynomial ring in more than one variables over an arbitrary algebraically closed field is not pole assignable that is one can find a pair of matrices over the ring which is completely reachable but not pole assignable

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

superconvergence for discontinuous galerkin finite element methods by l2 projection methods a general superconvergence of discontinuous galerkin dg finite element method for the elliptic problem is established by using l 2 projection method regularity assumptions for the elliptic problem with regular partitions are required numerical experiments are given to verify the theoretical results

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

stabilization of polynomially parametrized families of linear systems the single input case abstract given a continuous time family of finite dimensional single input linear systems parametrized polynomially such that each of the systems in the family is controllable there exists a polynomially parametrized control law making each of the systems in the family stable

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

superconvergence of h div finite element approximations for the stokes problem by l2 projection methods a general superconvergence result of h div finite element approximations for the stokes equations is established by using l 2 projection method regularity assumptions for the stokes problem with regular partitions is required numerical experiments are given to verify the theoretical results

enhancing finite element approximation for eigenvalue problems by projection method abstract this paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems the projection method is a postprocessing procedure that constructs a new approximation by using the least squares method the results are based on some regularity assumption for the elliptic problem and are applicable to the finite element approximations of self adjoint elliptic eigenvalue problems with general quasi regular partitions therefore the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes

a symmetric finite volume scheme for selfadjoint elliptic problems based on a linear finite element space a symmetric finite volume scheme for a self adjoint elliptic boundary value problem is proposed error estimates in l2 norm h1 norm and l norm are derived some post processing techniques are also provided

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

asymptotically exact a posteriori error estimators part i grids with superconvergence in part i of this work we develop superconvergence estimates for piecewise linear finite element approximations on quasi uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in particular we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant ui we then analyze a postprocessing gradient recovery scheme showing that q h nabla u h is a superconvergent approximation to nabla u here qh is the global l2 projection in part ii we analyze a superconvergent gradient recovery scheme for general unstructured shape regular triangulations this is the foundation for an a posteriori error estimate and local error indicators

can we have superconvergent gradient recovery under adaptive meshes we study adaptive finite element methods for elliptic problems with domain corner singularities our model problem is the two dimensional poisson equation results of this paper are twofold first we prove that there exists an adaptive mesh gauged by a discrete mesh density function under which the recovered gradient by the polynomial preserving recovery ppr is superconvergent second we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on ppr does produce adaptive meshes that asatisfy our mesh density assumption and the recovered gradient by ppr is indeed superconvergent in the adaptive process

adaptive finite element methods for hydrodynamic lubrication with cavitation we present an adaptive finite element method for a cavitation model based on reynolds equation a posteriori error estimates and adaptive algorithms are discussed and numerical examples illustrating the theory are supplied

local a posteriori estimates on a nonconvex polygonal domain a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems on a nonconvex polygonal domain the results are extensions of previous results for smooth domains w hoffmann et al math comp 70 2001 pp 897 909 a h schatz and l b wahlbin math comp 73 2004 pp 517 523

piecewise linear finite element methods are not localized recent results of schatz show that standard galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error these results do not indicate that such localization occurs when piecewise linear elements are used we show via simple one dimensional examples that schatz s estimates are sharp in that localization indeed does not occur when piecewise linear elements are used

localized pointwise error estimates for mixed finite element methods in this paper we give weighted or localized pointwise error estimates which are valid for two different mixed finite element methods for a general second order linear elliptic problem and for general choices of mixed elements for simplicial meshes these estimates similar in spirit to those recently proved by schatz for the basic galerkin finite element method for elliptic problems show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak this localization is more pronounced for higher order elements our estimates indicate that localization occurs except when the lowest order brezzi douglasmarini elements are used and we provide computational examples showing that the error is indeed not localized when these elements are employed

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second order quasilinear elliptic problems two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second order quasilinear elliptic dirichlet boundary value problems over convex polyhedral domains omega in space dimension n ge 2 we first give a residual estimator which is equivalent to nabla u u h l infty omega up to higher order terms the second type of residual estimator is designed to control nabla u u h locally over any subdomain of omega it is a novel a posteriori counterpart to the localized or weighted a priori estimates of sch98 this estimator is shown to be equivalent up to higher order terms to the error measured in a weighted global norm which depends on the subdomain of interest all estimates are proved for general shape regular meshes which may be highly graded and unstructured the constants in the estimates depend on the unknown solution u in the nonlinear case but in a fashion which places minimal restrictions on the regularity of u

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

superconvergence for the gradient of finite element approximations by l 2 projections a gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy the recovery technique is based on the method of least squares surface fitting in a finite dimensional space corresponding to a coarse mesh it is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces the recovery technique is robust efficient and applicable to a wide class of problems such as the stokes and elasticity equations

local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems we prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second order linear elliptic model problem first we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error the pollution term using a mesh dependent weight the residual term is bounded in a sharply localized fashion in specific situations the pollution term may also be bounded by computable residual estimators on nonconvex polygonal and polyhedral domains in two and three space dimensions we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks the finite element mesh is only required to be simpliciai and shape regular so that highly graded and unstructured meshes are allowed

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part ii the piecewise linear case we extend results from part i about estimating gradient errors elementwise a posteriori given there for quadratic and higher elements to the piecewise linear case the key to our new result is to consider certain technical estimates for differences in the error e x 1 e x 2 rather than for e x itself we also give a posteriori estimators for second derivatives on each element

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

local ultraconvergence of linear and bilinear finite element method for second order elliptic problems abstract in this article local ultraconvergence in gradient of the bilinear and linear finite element solutions to two dimensional second order elliptic problems has been investigated special interpolation postprocessing algorithms of numerical solutions by richardson extrapolation have been developed the local symmetry theory and estimates of discrete green functions are used in the analysis numerical experiments are provided to confirm the theoretical findings

recovery based error estimator for interface problems conforming linear elements this paper studies a new recovery based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems instead of recovering the gradient in the continuous finite element space the flux is recovered through a weighted l 2 projection onto h mathrm div conforming finite element spaces the resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results this paper also proposes an adaptive finite element method based on either the recovery based estimators or the edge estimator through local mesh refinement and establishes its convergence in particular it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

an adaptive finite element method for the inequality constrained reynolds equation abstract we present a stabilized finite element method for the numerical solution of cavitation in lubrication modeled as an inequality constrained reynolds equation the cavitation model is written as a variable coefficient saddle point problem and approximated by a residual based stabilized method based on our recent results on the classical obstacle problem we present optimal a priori estimates and derive novel a posteriori error estimators the method is implemented as a nitsche type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods

a posteriori error estimator based on gradient recovery by averaging for discontinuous galerkin methods we consider some anisotropic and piecewise constant diffusion problems in domains of r 2 approximated by a discontinuous galerkin method with polynomials of any fixed degree we propose an a posteriori error estimator based on gradient recovery by averaging it is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability the lower bound is also established moreover these additional terms are negligible when the recovered gradient is superconvergent the reliability and efficiency of the proposed estimator is confirmed by some numerical tests

improved zz a posteriori error estimators for diffusion problems conforming linear elements abstract in cai and zhang 2009 we introduced and analyzed an improved zienkiewicz zhu zz estimator for the conforming linear finite element approximation to elliptic interface problems the estimator is based on the piecewise constant flux recovery in the h div conforming finite element space this paper extends the results of cai and zhang 2009 to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear h div space

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

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

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

parametric stabilization is easy a polynomially parametrized family of continuous time controllable linear systems is always stabilizabte by polynomially parametrized feedback

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

pointwise error estimates for finite element solutions of the stokes problem in this paper pointwise error estimates for general finite element approximations of the stokes problem are established on quasi uniform grids in r n the results obtained in this paper improve and extend the existing error estimates in the maximum norm for the stokes problem the new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by product provide logarithm free bounds for all errors except the error of the velocity approximation of the lowest order

asymptotically exact a posteriori error estimators part ii general unstructured grids in part i of this work siam j numer anal 41 2003 pp 2294 2312 we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in this work we consider superconvergence for general unstructured but shape regular meshes we develop a postprocessing gradient recovery scheme for the finite element solution uh inspired in part by the smoothing iteration of the multigrid method this recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact next we use the superconvergent gradient to approximate the hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms we provide several numerical examples illustrating the effectiveness of our procedures

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

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

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