robust a posteriori error estimation for finite element approximation to h curl problem abstract in this paper we introduce a novel a posteriori error estimator for the conforming finite element approximation to the h curl problem with inhomogeneous media and with the right hand side only in l 2 the estimator is of the recovery type independent with the current approximation to the primary variable the electric field an auxiliary variable the magnetizing field is recovered in parallel by solving a similar h curl problem an alternate way of recovery is presented as well by localizing of the error flux the estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable it is proved that the estimator is approximately equal to the true error in the energy norm without the quasi monotonicity assumption finally we present numerical results for several h curl interface problems

robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems we present a patch based equilibrated flux recovery procedure for the conforming finite element approximation to diffusion problems the recovered flux is computed as the solution to a local constraint free minimization problem on each patch the approach is valid for higher order conforming elements in both two and three dimensions the resulting estimator admits guaranteed reliability and the robust local efficiency is proved under the quasi monotonicity condition of the diffusion coefficient numerical experiments are given to confirm the theoretical results

large character sums pretentious characters and the polya vinogradov theorem in 1918 polya and vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate one of the main results of this paper provides a substantial improvement of the polya vinogradov bound for characters of odd bounded order in 1977 montgomery and vaughan showed how the polya vinogradov inequality may be sharpened assuming the generalized riemann hypothesis we give a simple proof of their estimate and provide an improvement for characters of odd bounded order the paper also gives characterizations of the characters for which the maximal character sum is large and it finds a hidden structure among these characters

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

an uncertainty principle for arithmetic sequences analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are well distributed in some appropriate sense in various discrepancy problems combinatorics researchers have analyzed limitations to equidistribution as have fourier analysts when working with the uncertainty principle in this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory formulating a general principle and giving several examples

gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes in this paper the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes both the global and the local a posteriori error estimates are derived moreover it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth

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

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

superconvergent derivative recovery for lagrange triangular elements of degree p on unstructured grids in this paper we develop a postprocessing derivative recovery scheme for the finite element solution u h on general unstructured but shape regular triangulations in the case of continuous piecewise polynomials of degree p geq 1 by applying the global l 2 projection q h and a smoothing operator s h the recovered p th derivatives s h m q h partial p u h superconverge to the exact derivatives partial p u based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the p 1 st derivatives approximated by partial s h m q h partial p u h we provide several numerical examples illustrating the effectiveness of our schemes we also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence

maximum principle for mean field jump diffusion stochastic delay differential equations and its application to finance abstract this paper investigates a stochastic optimal control problem with delay and of mean field type where the controlled state process is governed by a mean field jump diffusion stochastic delay differential equation two sufficient maximum principles and one necessary maximum principle are established for the underlying system as an application a bicriteria mean variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques under certain conditions explicit expressions are provided for the efficient portfolio and the efficient frontier which are as elegant as those in the classical mean variance problem without delays

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

flux recovery and a posteriori error estimators conforming elements for scalar elliptic equations in this paper we first study two flux recovery procedures for the conforming finite element approximation to general second order elliptic partial differential equations one is accurate in a weighted l 2 norm studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 for linear elements and the other is accurate in a weighted h mathrm div norm up to the accuracy of the current finite element approximation for the l 2 recovered flux we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual based estimator based on the h mathrm div recovered flux we introduce two a posteriori error estimators one estimator may be regarded as an extension of the recovery based estimator studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to higher order conforming elements the global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection nor reaction dominant the other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem numerical results on test problems for these estimators are also presented

completely multiplicative functions taking values in 1 1 define the liouville function for a a subset of the primes p by a n 1 a n where a n is the number of prime factors of n coming from a counting multiplicity for the traditional liouville function a is the set of all primes denote l a x n a n and r a l i m l a n n n n 0 and 3 3 1 for the partial sums of character like functions we give exact values and asymptotics in particular we prove the following theorem theorem if p is an odd prime then max p k logx n x k x this result is related to a question of erd s concerning the existence of bounds for number theoretic functions within the course of discussion the ratio n n is considered

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

a hybrid a posteriori error estimator for conforming finite element approximations abstract this paper introduces a hybrid a posteriori error estimator for the conforming finite element method which may be regarded as a combination of the explicit residual and the improved zz error estimators with comparable cost the hybrid estimator is more accurate than the residual estimator it is shown that the hybrid estimator is reliable on all meshes unlike estimators of the zz type moreover the reliability constant is independent of the jump of the diffusion coefficients for elliptic interface problems under the monotonicity assumption of the coefficients finally numerical examples confirm the robustness of the estimator with respect to coefficient jumps and also better effectivity index compared to the residual estimator

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

some new residual based a posteriori error estimators for the mortar finite element methods in this paper we propose some new residual based a posteriori error estimators for the mortar finite element discretization of the second order elliptic equations with discontinuous coefficients reliability and efficiency of the estimators are given our analysis does not require any saturation assumptions and the mesh restrictions on the interface which are often needed in the literature numerical experiments are presented to confirm our theoretical analysis

stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of hamilton jacobi bellman equations abstract in this paper we study stochastic optimal control problems with jumps with the help of the theory of backward stochastic differential equations bsdes with jumps we generalize the results of peng s peng bsde and stochastic optimizations in j yan s peng s fang l wu topics in stochastic analysis science press beijing 1997 chapter 2 in chinese by considering cost functionals defined by controlled bsdes with jumps the application of bsde methods in particular the use of the notion of stochastic backward semigroups introduced by peng in the above mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps we prove that the value functions are the viscosity solutions of the associated generalized hamilton jacobi bellman equations with integral differential operators for this proof we adapt peng s bsde approach given in the above mentioned reference developed in the framework of stochastic control problems driven by brownian motion to that of stochastic control problems driven by brownian motion and poisson random measure

recovery based error estimator for stabilized finite element methods for the stokes equation abstract a recovery based error estimator is proposed and analyzed for stabilized p 1 p 0 continuous linear velocity constant pressure finite element approximations to the stokes equation reliability and efficiency of the estimator are established for various stabilized methods for several test problems numerical results show that our estimator is more accurate than the classical residual error estimator

a recovery based error estimate for nonconforming finite volume methods of interface problems in cai and zhang 2009 2010 12 13 they introduced the recovery based a posteriori error estimator for conforming mixed and nonconforming finite element methods of interface problems in this paper we extend the idea to present a recovery based a posterior error estimator for finite volume methods which employ the nonconforming linear trial functions to approximate elliptic interface problems the method recovers the flux and gradient in h div and h curl conforming finite element spaces with a weighted l 2 projection respectively the reliability and efficiency bounds are established numerical experiments are given to support the conclusions

a posteriori error analysis of the crank nicolson finite element method for linear parabolic interface problems a reconstruction approach abstract this article studies a residual based a posteriori error analysis for the crank nicolson time stepping finite element method for a linear parabolic interface problem in a bounded convex polygonal domain in r 2 a piecewise linear finite element space is used in space that is allowed to change in time and a modified crank nicolson approximation is applied for the time discretizations we employ a space time reconstruction that is piecewise quadratic in time and the clement type interpolation estimates to derive optimal order in time and an almost optimal order in space a posteriori error bound in the l l 2 norm the interface is assumed to be of arbitrary shape but is of class c 2 for our purpose numerical results are presented to validate the derived estimators

a posteriori error estimates for the stabilization of low order mixed finite elements for the stokes problem abstract this paper studies a posteriori error estimates for the stabilization of low order mixed finite elements for the stokes problem an interesting property of stabilized p 1 p 0 and p 1 p 1 finite element methods is proposed and used to construct new reliable and efficient error estimators moreover an average technique is applied to improve the error estimators numerical results verify the theoretical results and show the improvement of such an average technique

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

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

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

analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method ii abstract this is the first in a series of two papers in which the patch recovery technique proposed by zienkiewicz and zhu is analyzed in this work it is proved that the recovered derivative by the least squares fitting is superconvergent for the two point boundary value problems further the a posteriori error estimator based on the recovery technique is shown to be asymptotically exact this confirms the computational results obtained in 1 2

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

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

two grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data abstract in this paper we propose and analyze a two grid finite element method for a class of quasilinear elliptic problems under minimal regularity of data in a bounded convex polygonal r 2 which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space with this technique solving a quasilinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasilinear elliptic problem on a coarse space convergence estimates in the h 1 norm are derived to justify the efficiency of the proposed two grid algorithm moreover we propose a natural and computationally efficient residual based a posteriori error estimator of the two grid finite element method for this nonmonotone quasilinear elliptic problem and derive the global upper and lower bounds on the error in the h 1 norm numerical experiments are provided to confirm our theoretical findings

infinite horizon optimal control of forward backward stochastic differential equations with delay we consider a problem of optimal control of an infinite horizon system governed by forward backward stochastic differential equations with delay sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived we illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay

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

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

a recovery based a posteriori error estimator for h curl interface problems abstract this paper introduces a new recovery based a posteriori error estimator for the lowest order nedelec finite element approximation to the h curl interface problem the error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results under certain assumptions it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients

maximum principle for infinite horizon optimal control of mean field backward stochastic systems with delay and noisy memory in this paper we consider a problem of optimal control of an infinite horizon mean field backward stochastic differential equation with delay and noisy memory under partial information we derive

on sharp transitions in making squares in the fastest performing integer factoring algorithms one creates a sequence of integers in a pseudo random way and wishes to rapidly determine a subsequence whose product is a square in 1994 pomerance stated the following problem which encapsulates all of the key issues select integers a1 a2 at random from the interval 1 x until some nonempty subsequence has product equal to a square find a good estimate for the expected stopping time of this process a good solution should allow one to determine the optimal choice of parameters in many factoring algorithms pomerance 1994 using an idea of schroeppel 1985 showed that with probability 1 o 1 the rst subsequence whose product equals a square

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

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

rigidity theorems for multiplicative functions we establish several results concerning the expected general phenomenon that given a multiplicative function f mathbb n rightarrow mathbb c the values of f n and f n a are generally independent unless f is of a special form first we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values this implies the solution of the following folklore conjecture for any completely multiplicative function f mathbb n rightarrow mathbb t we have begin aligned liminf n rightarrow infty f n 1 f n 0 end aligned second we settle an old conjecture due to chudakov on the generalized characters in actes du congres international des mathematiciens nice 1970 tome 1 p 487 gauthier villars paris that states that any completely multiplicative function f mathbb n rightarrow mathbb c that a takes only finitely many values b vanishes at only finitely many primes and c has bounded discrepancy is a dirichlet character this generalizes previous work of tao on the erd s discrepancy problem finally we show that if many of the binary correlations of a 1 bounded multiplicative function are asymptotically equal to those of a dirichlet character chi mod q then f n chi n n it for all n where chi is a dirichlet character modulo q and t in mathbb r this establishes a variant of a conjecture of h cohn for multiplicative arithmetic functions the main ingredients include the work of tao on logarithmic elliott conjecture correlation formulas for pretentious multiplicative functions developed earlier by the first author and szemeredi s theorem for long arithmetic progressions

on the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients the successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients an a posteriori error indicator a refinement coarsening strategy and the choice of various parameters the objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem the linear elliptic equation with strongly discontinuous coefficients we derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm

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 convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

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

maximum principles for jump diffusion processes with infinite horizon abstract we prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information the results are applied to an optimal consumption and portfolio problem in infinite horizon

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

recovery based error estimators for interface problems mixed and nonconforming finite elements in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 we introduced and analyzed a recovery based a posteriori error estimator for conforming linear finite element approximation to interface problems it was shown theoretically that the estimator is robust with respect to the size of jumps provided that the distribution of coefficients is locally monotone numerical examples showed that this condition is unnecessary this paper extends the idea in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to mixed and nonconforming finite element methods for developing and analyzing robust estimators numerical results on test problems are also presented moreover an a priori error estimate is obtained when the underlying problem has low regularity

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