characterizations of locally testable linear and affine invariant families the linear or affine invariance is the property of a function family that is closed under linear or affine transformations on the domain and closed under linear combinations of functions respectively both the linear and affine invariant families of functions are generalizations of many symmetric families for instance the low degree polynomials kaufman and sudan 2007 21 introduced the notions of constraint and characterization to characterize the locally testable affine and linear invariant families of functions over finite fields of constant size in this article it is shown that for any finite field f of size q and characteristic p and its arbitrary extension field k of size q if an affine invariant family k n f has a k local constraint then it is k locally testable for k k 2 q pq 2 q p 4 and that if a linear invariant family k n f has a k local characterization then it is k locally testable for k 2k 2 q pq 4 q p 1 consequently for any prime field f of size q any positive integer k we have that for any affine invariant family over field f the four notions of the constraint the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities and that for any linear invariant family the notions of the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities the results significantly improve and are in contrast to the characterizations in 21 which have locality exponential in q even if the field k is prime in the research above a missing result is a characterization of linear invariant function families by the more natural notion of constraint for this we show that a single strong local constraint is sufficient to characterize the local testability of a linear invariant boolean function family and that for any finite field f of size q greater than 2 there exists a linear invariant function family over f such that it has a strong 2 local constraint but is not q d q 1 1 locally testable the proof for this result provides an appealing approach toward more negative results in the theme of characterization of locally testable algebraic properties which is rare and of course significant

three dimensional vibration analysis of circular and annular plates via the chebyshev ritz method abstract a three dimensional free vibration analysis of circular and annular plates is presented via the chebyshev ritz method the solution procedure is based on the linear small strain three dimensional elasticity theory selecting chebyshev polynomials which can be expressed in terms of cosine functions as the admissible functions a convenient governing eigenvalue equation can be derived through the ritz method according to the geometric properties of circular and annular plates the vibration is divided into three distinct categories axisymmetric vibration torsional vibration and circumferential vibration each vibration category can be further subdivided into the antisymmetric and symmetric ones in the thickness direction convergence and comparison study demonstrated the high accuracy and efficiency of the present method the present approach shows a distinct advantage over some other ritz solutions in that stable numerical operation can be guaranteed even when a large number of admissible functions is employed therefore not only lower order but also higher order eigenfrequencies can be obtained by using sufficient terms of the chebyshev polynomials finally some valuable results for annular plates with one or both edges clamped are given and discussed in detail

a finite volume method for a two phase multicomponent polymer flooding multicomponent polymer flooding used in enhanced oil recovery is governed by a system of coupled non strictly hyperbolic conservation laws in the presence of gravity the flux functions need not be monotone and hence designing godunov type upwind schemes is difficult and computationally expensive to overcome this difficulty we use the basic idea of discontinuous flux to reduce the coupled system into an uncoupled system of scalar conservation laws with discontinuous coefficients for these scalar equations we use the dflu flux developed in 5 to construct a second order scheme the scheme is shown to satisfy a maximum principle and the performance of the scheme is shown on both one and two dimensional test problems

a posteriori error estimation for an augmented mixed primal method applied to sedimentation consolidation systems abstract in this paper we develop the a posteriori error analysis of an augmented mixed primal finite element method for the 2d and 3d versions of a stationary flow and transport coupled system typically encountered in sedimentation consolidation processes the governing equations consist in the brinkman problem with concentration dependent viscosity written in terms of cauchy pseudo stresses and bulk velocity of the mixture coupled with a nonlinear advection nonlinear diffusion equation describing the transport of the solids volume fraction we derive two efficient and reliable residual based a posteriori error estimators for a finite element scheme using raviart thomas spaces of order k for the stress approximation and continuous piecewise polynomials of degree k 1 for both velocity and concentration for the first estimator we make use of suitable ellipticity and inf sup conditions together with a helmholtz decomposition and the local approximation properties of the clement interpolant and raviart thomas operator to show its reliability whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle bubble and edge bubble functions next we analyse an alternative error estimator whose reliability can be proved without resorting to helmholtz decompositions finally we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed primal finite element method

godunov type methods for conservation laws with a flux function discontinuous in space scalar conservation laws with a flux function discontinuous in space are approximated using a godunov type method for which a convergence theorem is proved the case where the flux functions at the interface intersect is emphasized a very simple formula is given for the interface flux a numerical comparison between the godunov numerical flux and the upstream mobility flux is presented for two phase flow in porous media a consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration furthermore for regular solutions uniqueness has been shown

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

axisymmetric straining modes in the vibration of circular plates axisymmetric straining modes which are different from the axisymmetric flexural modes of vibrating circular plates and cannot be revealed by conventional plate theory based approaches are investigated in the present study mode shapes of an example problem with clamped and four different simply supported boundary conditions are presented to show what straining modes may look like the straining modes are classified into three types and a criterion is set up to predict the appearance of straining modes some observations about straining modes are obtained and discussions are made results for different poisson ratios are also shown

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

finite element analysis of three dimensional vibrations of thick circular and annular plates abstract three dimensional vibrations of thick circular and annular plates are analyzed by a finite element method which with a properly assumed set of displacement field is different from the traditional 3 d finite element analysis and is reduced to a sequence of 2 d analyses one for each circumferential wave number the present approach has several unique features 1 it can obtain vibration frequencies which are comparable to or as accurate as those by other three dimensional approaches whenever comparisons between them are possible and yet the formulation of the present method is simpler and its application is straightforward 2 different simply supported boundary conditions can be exactly imposed with the present approach the latter feature may be quite difficult if not impossible for other methods in the present analysis vibration frequencies of a circular and an annular plate under different combinations of boundary conditions wave numbers and finite element meshes are calculated vibration characteristics are examined and some typical mode shapes are shown graphically then the results of some example problems by the present method are compared to those available in the literature to show its validity and accuracy

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

a fully discrete stabilized mixed finite volume element formulation for the non stationary conduction convection problem abstract at first a semi discrete formulation with respect to time for the non stationary conduction convection problem is recalled then a fully discrete stabilized mixed finite volume element smfve formulation based on two local gauss integrals and parameter free is established directly from the semi discrete formulation with respect to time for the non stationary conduction convection problem following this the error estimates for the fully discrete smfve solutions are derived by means of the standard mixed finite element method finally some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results the degrees of freedom of the fully discrete smfve formulation are far fewer than those of the classical finite volume element fve formulation without any stabilization and its numerical solutions are more stable than those of the classical fve formulation without any stabilization thus validating that the fully discrete smfve formulation is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem

checking approximate computations of polynomials and functional equations a majority of the results on self testing and correcting deal with programs which purport to compute the correct results precisely we relax this notion of correctness and show how to check programs that compute only a numerical approximation to the correct answer the types of programs that we deal with are those computing polynomials and functions defined by certain types of functional equations we present results showing how to perform approximate checking self testing and self correcting of polynomials settling in the affirmative a question raised by p gemmell et al proceedings of the 23rd acm symposium on theory of computing 1991 pp 32 42 r rubinfeld and m sudan proceedings of the third annual acm siam symposium on discrete algorithms orlando fl 1992 pp 23 43 r rubinfeld and m sudan siam j comput 25 1996 pp 252 271 we obtain this by first building approximate self testers for linear and multilinear functions we then show how to perform approximate checking self testing and self correcting for those functions that satisfy addition theorems settling a question raised by r rubinfeld siam j comput 28 1999 pp 1972 1997 in both cases we show that the properties used to test programs for these functions are both robust in the approximate sense and stable finally we explore the use of reductions between functional equations in the context of approximate self testing our results have implications for the stability theory of functional equations

algebraic testing and weight distributions of codes we study the testing problem that is the problem of determining maybe probabilistically if a function to which one has oracle access satisfies a given property we propose a framework in which to formulate and carry out the analysis of several known tests this framework establishes a connection between testing and the theory of weight distributions of codes we illustrate this connection by giving a coding theoretic interpretation of several tests that fall under the label of low degree tests we also show how the connection naturally suggests a new way of testing for linearity over finite fields we derive from the macwilliams theorems a general result the duality testing lemma and use it to analyze the simpler tests that fall into our framework in contrast to other analyses of tests the ones we present elicit the fact that a test s probability of rejecting a fuction depends on how far away the function is from every function that satisfies the property of interest

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

error estimates for discontinuous galerkin method for nonlinear parabolic equations abstract we consider the nonlinear parabolic partial differential equations we construct a discontinuous galerkin approximation using a penalty term and obtain an optimal l l 2 error estimate

a discontinuous mixed covolume method for elliptic problems we develop a discontinuous mixed covolume method for elliptic problems on triangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems in this paper we have analyzed a one parameter family of hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems nabla cdot rm a u nabla u f u nabla u 0 with dirichlet boundary conditions these methods depend on the values of the parameter theta in 1 1 where 1 corresponds to the nonsymmetric and 1 corresponds to the symmetric interior penalty methods when rm a u nabla u nabla u and f u u f that is for the poisson problem the error estimate in the broken h 1 norm which is optimal in h mesh size and suboptimal in p degree of approximation is derived using piecewise polynomials of degree p 2 when the solution u in h 5 2 omega in the case of linear elliptic problems also this estimate is optimal in h and suboptimal in p further optimal error estimate in the l 2 norm when 1 is derived numerical experiments are presented to illustrate the theoretical results

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

an augmented lagrangian method for the phase field approach for pressurized fractures abstract in the modeling of pressurized fractures using phase field approaches the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase field function in comparison to the classical case in elasticity the presence of the pressure requires the additional constraint and makes the problem much harder to analyze after temporal discretization this induces a minimization problem in each time step over a solution dependent admissible set to avoid solving the resulting variational inequality corresponding to the first order necessary conditions a penalization approach is used commonly to remove the inequality constraint it is well known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process consequently to avoid such a drawback we propose an augmented lagrangian algorithm for the discrete in time and continuous in space phase field problems the final set of equations is solved in a decoupled fashion the proposed method is substantiated with several benchmark and prototype tests in two and three dimensions

a stabilized finite volume element formulation for sedimentation consolidation processes a model of sedimentation consolidation processes in so called clarifier thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the stokes system for an incompressible fluid describing the motion of the mixture in cylindrical coordinates and if an axially symmetric solution is assumed the original problem reduces to two space dimensions this poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted sobolev spaces a novel finite volume element method is introduced for the spatial discretization where the velocity field and the solids concentration are discretized on two different dual meshes the method is based on a stabilized discontinuous galerkin formulation for the concentration field and a multiscale stabilized pair of mathbb p 1 mathbb p 1 elements for velocity and pressure respectively numerical experiments illustrate properties of the model and the satisfactory p

discontinuous finite volume element discretization for coupled flow transport problems arising in models of sedimentation the sedimentation consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the stokes problem written in terms of the mixture flow velocity and the pressure field here both the viscosity and the forcing term depend on the local solids concentration a semi discrete discontinuous finite volume element dfve scheme is proposed for this model the numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field whereas the pressure is approximated by piecewise constant elements the unique solvability of both the nonlinear continuous problem and the semi discrete dfve scheme is discussed and optimal convergence estimates in several spatial norms are derived properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples including flows under gravity with changing direction a secondary settling tank in an axisymmetric setting and batch sedimentation in a tilted cylindrical vessel

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

axisymmetric vibration analysis of rotating annular plates by a 3d finite element the axisymmetric vibration of spinning isotropic annular plates is herein analyzed by a modified axisymmetric finite element due to the three dimensional elasticity nature of the present approach all the initial stresses induced by the centrifugal force are calculated and taken into account in the augmented strain energy expression while only two are considered in the conventional analyses also the whole three velocity components are considered in the inertial term instead of only one by conventional approaches natural frequencies of annular plates with different ratios of inner to outer radius ratios of thickness to radius rotational speeds and boundary conditions are derived by the present approach and are compared with those by a plate theory approach

a discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

a limiting approach for dg discretizations on mixed type meshes abstract a limiting procedure of discontinuous galerkin dg discretizations that is suitable for computations of high speed flows with strong shocks around complex geometries using a p adaptive procedure on mixed type quadrilateral triangular meshes is developed a total variation bounded tvb approach is the building block of the limiting procedure the main difference of this work with other recent limiting approaches for dg discretizations that were based on tvb limiting is that a unified approach for limiting on mixed type meshes is obtained this is achieved by applying the tvb limiter on the transformed canonical elements where appropriate estimates of the variation of the approximate solution derivatives are feasible applications for mixed type meshes and p adaptive calculations with the dg method are performed for a number of standard test cases for shock capturing schemes to illustrate the potential of the proposed limiting approach it is demonstrated in the computed results that good resolution of discontinuities and embedded smooth flow features is obtained with quadrilateral triangular and mixed type meshes furthermore the resolution of the present approach on arbitrary type meshes is equally good with the resolution obtained with other limiting approaches of equivalent order obtained on rectangular meshes

a new discontinuous finite volume method for elliptic problems we develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients an optimal order error estimate is obtained in a mesh dependent norm an l2 error estimate is also obtained

on the robustness of functional equations in this paper we study the general question of how characteristics of functional equations influence whether or not they are robust we isolate examples of properties which are necessary for the functional equations to be robust on the other hand we show other properties which are sufficient for robustness we then study a general class of functional equations which are of the form forall x y f f x y f x y f x f y 0 where f is an algebraic function we give conditions on such functional equations that imply robustness our results have applications to the area of self testing correcting programs we show that self testers and self correctors can be found for many functions satisfying robust functional equations including algebraic functions of trigonometric functions such as tan x 1 over 1 cot x ax over 1 ax cosh x

mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media abstract in this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media the proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise then discretise approach for the approximation of the optimal control problem leading to nonsymmetric algebraic systems and employing minimum regularity requirements estimates for the error between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed discontinuous approximation measured in suitable norms are derived showing optimal orders of convergence

partitioned coupling of advection diffusion reaction systems and brinkman flows we present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection diffusion reaction systems and incompressible viscous flow the space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction advection and diffusion processes considering the global system as a number of sequentially linked sets of partial differential and algebraic equations the flow solver presents the advantage that all unknowns in the system here vorticity velocity and pressure can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective the robustness of the proposed method is illustrated with a series of numerical tests in 2d and 3d relevant in the modelling of bacterial bioconvection and boussinesq systems

new fully mixed finite element methods for the stokes darcy coupling abstract in this paper we introduce and analyze two new fully mixed variational formulations for the coupling of fluid flow with porous media flow flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we first extend recent related results involving a pseudostress velocity based formulation in the fluid and consider a fully mixed formulation in which the main unknowns are given now by the stress the vorticity and the velocity all them in the fluid together with the velocity and the pressure in the porous medium the aforementioned formulation is then partially augmented by introducing galerkin least squares type terms arising from the constitutive and equilibrium equations of the stokes equation and from the relation defining the vorticity in terms of the free fluid velocity these three terms are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well posed the classical babuska brezzi theory is applied to provide sufficient conditions for the well posedness of the continuous and discrete formulations of both approaches next we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element scheme the proof of reliability makes use of the global inf sup condition helmholtz decomposition and local approximation properties of the clement interpolant and raviart thomas operator in turn inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools to prove the efficiency of the estimator finally several numerical results illustrating the good performance of both methods confirming the aforementioned properties of the estimator and showing the behavior of the associated adaptive algorithm are provided