the adaptive computation of far field patterns by a posteriori error estimation of linear functionals this paper is concerned with the derivation of a priori and a posteriori error bounds for a class of linear functionals arising in electromagnetics which represent the far field pattern of the scattered electromagnetic field the a posteriori error bound is implemented into an adaptive finite element algorithm and a series of numerical experiments is presented

a posteriori error analysis for stabilised finite element approximations of transport problems we develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments two alternative dual problems are considered one is based on the formal adjoint of the hyperbolic differential operator the other on the transposition of the bilinear form for the stabilised finite element method we show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes

spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

sequential quantum measurements a quantum effect is an operator a on a complex hilbert space h that satisfies 0 a i we denote the set of quantum effects by e h the set of self adjoint projection operators on h corresponds to sharp effects and is denoted by p h we define the sequential product of a b e h by a b a1 2ba1 2 the main purpose of this article is to study some of the algebraic properties of the sequential product many of our results show that algebraic conditions on a b imply that a and b commute for the usual operator product for example if a b satisfies certain distributive or associative laws then ab ba moreover if a b p h then ab ba and a b b or b a b if and only if ab ba b a natural definition of stochastic independence is introduced and briefly studied

holonomy rigidity for ricci flat metrics on a closed connected oriented manifold m we study the space mathcal m vert m of all riemannian metrics which admit a non zero parallel spinor on the universal covering such metrics are ricci flat and all known ricci flat metrics are of this form we show the following the space mathcal m vert m is a smooth submanifold of the space of all metrics and its premoduli space is a smooth finite dimensional manifold the holonomy group is locally constant on mathcal m vert m if m is spin then the dimension of the space of parallel spinors is a locally constant function on mathcal m vert m

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

a spinorial energy functional critical points and gradient flow let m be a compact spin manifold on the universal bundle of unit spinors we study a natural energy functional whose critical points if dim m ge 3 are precisely the pairs g varphi consisting of a ricci flat riemannian metric g together with a parallel g spinor varphi we investigate the basic properties of this functional and study its negative gradient flow the so called spinor flow in particular we prove short time existence and uniqueness for this flow

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

on a posteriori pointwise error estimation using adjoint temperature and lagrange remainder the pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed the truncation error is estimated using a taylor series with the remainder in the lagrange form the contribution of the local error to the total pointwise error is estimated via an adjoint temperature it is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found

on the full holonomy group of lorentzian manifolds the classification of restricted holonomy groups of n dimensional lorentzian manifolds was obtained about ten years ago however up to now not much is known about the structure of the full holonomy group in this paper we study the full holonomy group of lorentzian manifolds with a parallel null line bundle based on the classification of the restricted holonomy groups of such manifolds we prove several structure results about the full holonomy we establish a construction method for manifolds with disconnected holonomy starting from a riemannian manifold and a properly discontinuous group of isometries this leads to a variety of examples most of them being quotients of pp waves with disconnected holonomy including a non flat lorentzian manifold with infinitely generated holonomy group furthermore we classify the full holonomy groups of solvable lorentzian symmetric spaces and of lorentzian manifolds with a parallel null spinor finally we construct examples of globally hyperbolic manifolds with complete spacelike cauchy hypersurfaces disconnected full holonomy and a parallel spinor

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

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

holonomy groups of compact riemannian manifolds a classification in dimensions up to ten the possible holonomy groups of compact locally irreducible riemannian manifolds are studied motivated by applications arising in physics it is not required that these manifolds be simply connected in this paper a complete classification up to dimension ten is given with particular emphasis on the manifolds of dimension six or less in this latter case there exist actual examples for every class so that the possible holonomy groups can be characterized exactly

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

adjoint correction and bounding of error using lagrange form of truncation term the a posteriori error evaluation based on differential approximation of a finite difference scheme and adjoint equations is addressed the differential approximation is composed of primal equations and a local truncation error determined by a taylor series in lagrange form this approach provides the feasibility of both refining the solution and using the holder inequality for asymptotic bounding of the remaining error

methods of holonomy theory for ricci flat riemannian manifolds compact ricci flat riemannian manifolds often arise in physical applications either as a technical device or as models of internal space the idea of extending the holonomy group of such a manifold to a larger gauge group embedding the connection in the gauge group plays a fundamental role in the manifold compactification approach to superstring phenomenology and the work of gepner suggests that this idea may have equally fundamental analogs in other approaches the holonomy theory of simply connected ricci flat manifolds has recently been the subject of much mathematical work but physicists are mainly interested in the case of multiply connected manifolds the purpose of this paper is to present some techniques for understanding the holonomy theory of compact multiply connected ricci flat manifolds these lead to a general classification theorem

the quotient construction for a class of compact einstein manifolds given any einstein manifoldm e one can obtain further examples of einstein manifolds by taking the quotientm e g by a freely acting properly discontinuous group of isometries we study this method in the case in whichm e is kahlerian m e g is compact and the ricci curvature is non negative in many cases the candidates forg can be completely classified

extensions of goal oriented error estimation methods to simulations of highly nonlinear response of shock loaded elastomer reinforced structures this paper describes extensions of goal oriented methods for a posteriori error estimation and control of numerical approximation to a class of highly nonlinear problems in computational solid mechanics an updated lagrangian formulation of the dynamical large deformation response of structures composed of strain rate sensitive elastomers and elastoplastic materials is developed to apply the theory of goal oriented error estimation a backward in time dual formulation of these problems is derived and residual error estimators for meaningful quantities of interest are established the target problem class is that of axisymmetric deformations of layered elastomer reinforced shells of revolution subjected to shock loading extensive numerical results on solutions of representative problems are given it is shown that extensions of the theory of goal oriented error estimation can be developed and applied effectively to a class of highly nonlinear multi physics problems in solid and structural mechanics

differentiability of the objective in a class of coefficient inverse problems abstract in this paper we study the frechet differentiability of the objective functional for a quite general class of coefficient inverse problems we present sufficient conditions for the existence of arbitrary order differentials as well as formulae for the calculation of first and second order derivatives the general case theorems are then applied to two real world geophysical inverse problems

a generalization of gudder nagy s theorem with numerical ranges of operators in this paper we prove that for a b c which are bounded linear operators on a separable complex hilbert space h ax x bx x cx x for all x h with x 1 if and only if there exists a complex number c c such that a ci and c cb or b ci and c ca

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

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

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

numerical ranges of the powers of an operator abstract the numerical range w a of a bounded linear operator a on a hilbert space is the collection of complex numbers of the form a v v with v ranging over the unit vectors in the hilbert space in terms of the location of w a inclusion regions are obtained for w a k for positive integers k and also for negative integers k if a 1 exists related inequalities on the numerical radius w a sup w a and the crawford number c a inf w a are deduced

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

a new goal oriented formulation of the finite element method abstract in this paper we introduce analyze and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary value problem the objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal oriented adaptivity which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement in the current approach we first solve the adjoint problem then use the adjoint information as a minimization constraint for the primal problem as a result the constrained finite element solution is enhanced with respect to the quantities of interest while maintaining near optimality in energy norm we describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples

grid adaptation for functional outputs application to two dimensional inviscid flows this paper presents an error estimation and grid adaptive strategy for estimating and reducing simulation errors in integral outputs functionals of partial differential equations adaptive criteria are derived using an adjoint based error correction technique that relates the local residual errors of both the primal and adjoint solutions to the global error in the prescribed functional this relationship allows local error contributions to be used as indicators in a grid adaptive strategy designed to produce specially tuned grids for accurately estimating the chosen functional in this paper attention is limited to two dimensional inviscid flows using a standard finite volume discretization although the procedure may be readily applied to other types of multidimensional problems and discretizations numerical examples demonstrate that the proposed adaptive procedure compares favorably in terms of accuracy efficiency and robustness relative to a commonly used gradient based adaptive method

examples of einstein manifolds with all possible holonomy groups in dimensions less than seven in an earlier work the possible holonomy groups of all compact locally irreducible riemannian manifolds of dimensions up to ten were classified placing particular emphasis on the non simply connected case in this work the problem of finding examples of manifolds with such holonomy groups is discussed it is proven in particular that it is possible to find einsteinian examples of every one of the 23 holonomy types corresponding to manifolds of dimensions less than 7 thus an exact characterization of the groups that can occur as holonomy groups can be given in those dimensions

a note on operator probability theory involving numerical ranges abstract in this note the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered

on error controlled numerical model reduction in fe2 analysis of transient heat flow numerical model reduction is exploited for solving the two scale problem that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the method of spectral decomposition a key feature is the construction of a symmetrized version of the space time variational format by which it is possible to estimate the error from the model reduction in i energy norm and in ii a given quantity of interest in previous work by the authors this strategy has been applied to the solution of an individual representative volume element problem whereas the error transport to the macroscale problem in the finite element squared fe2 approach was ignored in this paper such transport of error is included in the error estimate it is remarkable that it is still possible to obtain guaranteed bounds on the error as compared to the fully resolved discrete finite element problem while using only the reduced basis and with minor extra computational effort the performance of the error estimates is demonstrated via numerical results whereby the subscale is modeled in three dimensions whereas the macroscale problem is either one or two dimensional

parallel spinors and holonomy groups in this paper we complete the classification of spin manifolds admitting parallel spinors in terms of the riemannian holonomy groups more precisely we show that on a given n dimensional riemannian manifold spin structures with parallel spinors are in one to one correspondence with lifts to spinn of the riemannian holonomy group with fixed points on the spin representation space in particular we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

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

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

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

enhancing least squares finite element methods through a quantity of interest in this paper we introduce an approach that augments least squares finite element formulations with user specified quantities of interest the method incorporates the quantity of interest into the least squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity of interest we establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions central to the approach is that it offers an element level estimate of the error in the quantity of interest as a result we introduce an adaptive approach that yields efficient adaptively refined approximations several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology notably the results show that the new approach is effective at improving the accuracy per total computational cost

a new adaptive remeshing scheme based on the sensitivity analysis of the spr point wise error estimation abstract this paper presents a formulation for the obtainment of the sensitivity analysis of a point wise error estimator with respect to the nodal coordinates using the adjoint state method the proposed point wise error estimator is based on the spr method the numerical accuracy of the presented sensitivity analysis has been tested by perturbing a mesh the capability of the presented sensitivity analysis for the detection of pollution error has also been tested a new adaptive remeshing strategy based on the sensitivity analysis of the point wise error estimation has been developed and tested this strategy produces very cheap meshes for the accurate evaluation of stresses at specific points

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