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

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

on the absence of infinite ab percolation clusters in bipartite graphs the authors give a rigorous proof that infinite ab percolation does not occur for any parameter value in a certain class of bipartite planar graphs in particular infinite ab percolation cannot occur on the square lattice

essential embedding of cyclic modules in projectives let r be a ring and e e rr its injective envelope we show that if every simple right r module embeds in rr and every cyclic submodule of er is essentially embeddable in a projective module then rr has finite essential socle as a consequence we prove that if each finitely generated right r module is essentially embeddable in a projective module then r is a quasifrobenius ring we also obtain several other applications and among them a we answer affirmatively a question of al huzali jain and lopez permouth by showing that a right cep ring i e a ring r such that every cyclic right module is essentially embeddable in a projective module is always right artinian b we prove that if r is right fgf i e any finitely generated right r module embeds in a free module and right cs then r is quasi frobenius

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

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

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

infinite ab percolation clusters exist on the triangular lattice infinite ab percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values it was not previously known that infinite ab percolation occurred with positive probability on any planar lattice

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

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

estimations a posteriori d un schema de volumes finis pour un probleme non lineaire resumedans cet article nous developpons des estimations a posteriori de l erreur commise en approchant un probleme elliptique non lineaire par une methode de volumes finis mvf de type vertex centered les estimateurs obtenus mesurent le residu de l equation forte et l irregularite de la solution discrete qui se traduit par des sauts a travers les inter elelments une condition sur la fonction flux numerique sera imposee pour traiter la non conformite du probleme au cours de l evaluation de l estimation summary we derive a posteriori error estimates for vertex centered finite volume discretizations of a class of non linear elliptic pdes the error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pde s prinicipal part across the element boundaries the non conformity of the discretization is taken into account by a condition on the numerical flux function

recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact or reference error lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs this work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates this lower bound may also be used to assess the effectivity of the former estimate and to improve it

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

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

ab percolation on close packed graphs for a class of close packed graphs the ab percolation critical probability is equal to the classical site percolation critical probability of the related graph obtained by inserting next nearest neighbour bonds as a consequence infinite ab percolation occurs for many fully triangulated graphs

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

a uniqueness condition for gibbs measures with application to the 2 dimensional ising antiferromagnet a uniqueness condition for gibbs measures is given this condition is stated in terms of absence of a certain type of percolation involving two independent realisations this result can be applied in certain concrete situations by comparison with ordinary percolation in this way we prove that the ising antiferromagnet on a square lattice has a unique gibbs measure if 4 h 1 2ln pc 1 pc whereh denotes the external magnetic field the inverse temperature andpc the critical probability for site percolation on that lattice sincepc is larger than 1 2 this extends a result by dobrushin kolafa and shlosman whose proof was computer assisted

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

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

the influence and selection of subspaces for a posteriori error estimators the element residual method for a posteriori error estimation is analyzed for degree p finite element approximation on quadrilateral elements the influence of the choice of subspace used to solve the element residual problem is studied it is shown that the resulting estimators will be consistent or asymptotically exact for all p 1 if and only if the mesh is parallel moreover even if the mesh consists of rectangles then the estimators can be inconsistent when p 1 the results provide concrete guidelines for the selection of a posteriori error estimators and establish the limits of their performance in particular the use of the element residual method for high orders of approximation such as those arising in the h p version finite element method is vindicated the mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual problem the investigations reveal a deficiency in the formulation leading as it does to spurious modes in the true solution of the residual problem the recommended choice of subspaces may be viewed as being sufficient to guarantee that the spurious modes are filtered out from the approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes

critical probabilities for site and bond percolation models cc weak inequality p site p bond is strengthened to strict inequality for a cc broad category of graphs g including all the usual finite dimensional lattices in two and more dimensions the complementary inequality p site c bond 1 1 1 p is proved also where denotes the supremum of c the vertex degrees of g 0 introduction and results let g v e be an infinite connected graph our target in this paper is to study the relationship between site and bond percolation on g and particularly to prove inequalities between the two critical probabilities p site p bond of these models to date the only general cc inequality of this type appears to be the weak inequality p site p bond valid cc for all connected graphs g our principal purpose here is to extend the coupling arguments used in proving this weak inequality and to exploit recently developed methods for proving strict inequalities in order to obtain the inequality p site p bond for a certain broad category of graphs cc several difficulties arise in this program under two general headings first a method is required for utilizing the strict inequality methods of aizenman and grimmett 1991 in the nonstatic setting which occurs when studying stochastic couplings of site and bond percolation second there are graph theoretic complications in applying such techniques to general graphs we shall give two theorems about strict inequalities rather than one only our first such result is theorem 1 which applies specifically to hypercubic lattices in d 2 dimensions this will be proved using a relatively straightforward construction although this construction may in principle be ex tended with some combinatorial complications to certain other graphs on an ad hoc basis it does not seem to be easy to adapt it to a broad general class of graphs therefore we have separated out in theorem 2 our more general result the proof of which requires substantially deeper ideas than that of

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

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

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

critical probability bounds for two dimensional site percolation models the authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two dimensional lattices a technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that pc d or 0 5020 for the dice lattice d combining this method with kesten s duality result simplifies toth s derivation of the lower bound pc s or 0 5034 for the square lattice s the authors also present a technique for deriving upper bounds for bipartite graphs a technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice h pc h or 0 8079 and pc h or pc s the grouping technique is applied to the dice lattice to find the upper bound pc d or 0 7937

asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method in this paper we analyze an a posteriori error estimator based on the equilibrated residual method we prove that this estimator is asymptotically exact in the energy norm for regular solutions and o h 1 alpha alpha 0 meshes numerical examples are included to illustrate the theoretical results

a comparison of a posteriori error estimators for mixed finite element discretizations by raviart thomas elements we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations in particular we present and analyze four different kinds of error estimators a residual based estimator a hierarchical one error estimators relying on the solution of local subproblems and on a superconvergence result respectively finally we examine the relationship between the presented error estimators and compare their local components

rings whose cyclics are essentially embeddable in projective modules a ring r is quasi frobenius qf if and only if every right r module is embeddable in a projective module we call a ring r a right left cepring if each cyclic right left r module is essentially embeddable in a projective module examples of right cep rings include qf rings and right uniserial rings indeed r is a qf ring if and only if r is both a right and left cep ring s a right cep ring which is qf 3 is shown to be qf theorem 3 3 semiperfect cep rings and rings each of whose homomorphic images is a right cep ring are characterized in theorems 5 2 and 6 2 respectively the last section deals with split extensions of right uniserial rings as examples of cep rings

on continuous rings abstract we show that ifris a semiperfect ring with essential left socle andrl k kfor every small right idealkofr thenris right continuous accordingly some well known classes of rings such as dual rings and rings all of whose cyclic rightr modules are essentially embedded in projectives are shown to be continuous we also prove that a ringrhas a perfect duality if and only if the dual of every simple rightr module is simple andr ris a left and right cs module in sect 2 of the paper we provide a characterization for semiperfect right self injective rings in terms of the cs condition

exact a posteriori error analysis of the least squares finite element method1this work was supported by nsc under grant nsc87 2115 m 009 005 taiwan 1 a residual type a posteriori error estimator is presented for the least squares finite element method the estimator is proved to equal the exact error in a norm induced by some least squares functional the error indicator of each element is equal to the exact error norm restricted to the element as well in other words the estimator is perfectly effective and reliable for error control and for adaptive mesh refinement the exactness property requires virtually no assumptions on the regularity of the solution and on the finite element order in the approximation or in the estimation the least squares method is in a very general setting that applies to various linear boundary value problems such as the elliptic systems of first order and of even order and the mixed type partial differential equations numerical results are given to demonstrate the exactness