R. E. Bank and A. Weiser, 1985. Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44 pp. 283-301.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The equilibrium method for finding error estimators for the finite element method was introduced by P. Ladeveze in his doctoral thesis. A more accessible reference is [1] Later the approach was applied in a global setting by Kelly in for example [2] and in a local setting by Bank and Weiser in [3]. Most recently the approach was applied again in a local setting by Ainsworth and Oden in [4] 5] and [6] Common for all approaches so far has been the recovery of the error in an a priori given norm, generally the energy norm. In [7] and [8] Hugger presents a method for the pointwise recovery ....

....to solving all equations in (40) with e replaced by zero globally for the equilibrium functions. Since we have rejected the idea of solving global problems for the error (except for the special case mentioned in section 1) this approach must also be rejected. Bank and Weiser suggested in [3] a local approach which in our setting corresponds to solving one equation in (40) with e replaced by zero for the equilibrium functions on the internal edges of that element. These traces are then transferred to the neighboring elements using the definition of U eq . Then (40) is solved with e ....

Randolph E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Mathematics of Computation, 44(170):283--301, 1985.

....and is therefore referred to as an implicit method [2] When compared to the simpler less expensive a posteriori explicit methods, which only require residual evaluations (e.g. 3] implicit methods o er the potential for quantitative constant free bounds. The original implicit methods [8, 1, 4] were developed, for linear self adjoint problems, to provide bounds for the energy norm of the error. In reality however, it is not the error in the energy norm which is of interest, but the error in the quantities on which an engineering decision will be based for instance de ection or stress. ....

....system. The solution of this problem is known as the equilibration problem. From the physical point of view, these hybrid uxes represent tractions that must be applied so that the each H macroelement is in equilibrium when considered in isolation. We follow here the approach proposed in [8, 4, 1], which requires solving an indeterminate system at each vertex of the TH grid, the size of which is given by the number of edges that meet at the vertex. 5 Bound Procedure for an Arbitrary Output We consider now the evaluation of upper and lower bounds for an arbitrary output s h = S( h ) ....

R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44:283-301, 1985.

.... the convergence rate, of (say) trust region [9, 19] and line search [11, 20] quasi Newton techniques [28] In all earlier a posteriori error analysis techniques, either in implicit approaches the measure of the error is not related to the actual engineering outputs of interest (e.g. [15, 4, 2]) or in explicit approaches the error estimates for the engineering outputs of interest involve numerous undetermined or uncertain constants or functions (e.g. 5, 6, 27] in both cases, quantitative confirmation and hence both certainty and e#ciency is seriously compromised, and ....

....to the design process, and should lay the foundation for systemic application of a posteriori error control within the engineering context. Although our method provides a critical new capability, we are nevertheless indebted to earlier a posteriori implicit (Neumann subproblem) techniques [15, 4, 2] for several important conceptual and mathematical ingredients in particular duality theory and flux hybridization. The former, though not strictly necessary and even sometimes restrictive provides a derivational mechanism without which the requisite equations are very di#cult to ....

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R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations, Math. Comp., 44:170 (1985), pp. 283--301.

....and is therefore referred to as an implicit method [2] When compared to the simpler more inexpensive a posteriori explicit methods which only require residual evaluations (e.g. 3] implicit methods o er the potential for quantitative constant free bounds. The original implicit methods [9] 1] [4] were developed, for linear self adjoint problems, to provide bounds for the energy norm of the error. In reality however, it is not the error in the energy norm which is of interest, but the error in the quantities on which an engineering decision will be based, such as stresses, de ection etc. ....

....; which yields uH . Now we can solve for pH , b( v; pH ) l( v) a(uH ; v) 8 v 2 X : 17) The above equations represent a solvable but indeterminate system. The computation of an acceptable compatible solution is known as the equilibration problem. We follow here the approach proposed in [9,4,1], which involves the solution of an indeterminate system at each vertex of the TH grid, the size of which is given by the number of edges that meet at the vertex. Details of this procedure can be found in [2] There exist other approaches to obtaining implicit error bounds which circumvent the ....

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R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44:283-301, 1985.

....lower error bounds. Our estimator is reliable and e#cient and furthermore robust for nearly incompressible materials. In the last section we present two error estimators based on the solution of auxiliary local problems. These error estimators are similar to the ones developed by Bank and Weiser [4]. In order to deal with these problems and to obtain estimates with respect to the standard norms, we need the results on the stability of BDMS elements which we presented in [12] 2 The Linear Elasticity Problem Let # , d = 2, 3, be a bounded polygonal or polyhedral domain with boundary # = ....

R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations., Math. Comput., 44 (1985), pp. 283-- 301.

....AMS Subject Classification. Primary 65N30, Secondary 65N50, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25. 1. Introduction. A posteriori error estimates have become standard in modern engineering and scientific computation. There are two types of popular error estimators: the residual type (see, e.g. [2, 4]) and the recovery type (see, e.g. 21] The most representative recovery type error estimator is the Zienkiewicz Zhu error estimator, especially the estimator based on gradient patch recovery by local discrete least squares fitting [22, 23] The method is now widely used in engineering practice ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations, Math. Comp., 44 (1985), pp.283-301.

....be neglected, if the triangulation is fine enough. We note that for each triangulation there is an example where this term is dominant and the error estimator fails. 6. 2 Hierarchical error estimator Hierarchical basis error estimators are well known for standard conforming discretizations [8, 40, 43, 90]. They are based on a defect correction in a higher order space and a hierarchical two level splitting. An excellent overview of different techniques can be found in [22, 90] see also the references therein) There are basically two ways to obtain such an error estimator. Using the ideas of ....

....in [22, 90] see also the references therein) There are basically two ways to obtain such an error estimator. Using the ideas of Deuflhard, Leinen, Yserentant [40] the resulting continuous defect problem is first discretized and then localized. The second possibility follows Bank and Weiser [8], where the defect problem is first localized and then discretized. These concepts have been generalized to nonconforming discretizations, 58, 95] mixed Raviart Thomas discretizations, 1, 29, 57, 60] and the Stokes problem [9, 88] In this section, we present a hierarchical basis error ....

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R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283--301.

.... same was done for high frequency scattering problems, i.e. the full Maxwell s equations, in [30] To the knowledge of the authors, hierarchical error estimator for quasistatic electromagnetics problems have not been investigated before, though they are widely used for ordinary elliptic problems [5, 21, 24, 32]. The paper is organized as follows: In the next section we introduce edge elements of lowest and second order. Then we discuss the principles of hierarchical error estima3 tion and present the concrete hierarchical error estimator. It turns out that it is essential to pay special attention to ....

R. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283--301.

....preconditioners see, e.g. 10, 11, 13] 3. Hierarchical type a posteriori error estimator For standard conforming finite element approximations of elliptic boundary value problems there are two different concepts for hierarchical type a posteriori error estimators due to Bank and Weiser [3] and Deuflhard, Leinen and Yserentant [9] Here, in case of mortar finite elements we first follow the ideas from [3] in so far as we consider a localization of the error equation on the subdomains level. Then, we proceed in much the same way as in [9] by approximating the individual error ....

.... finite element approximations of elliptic boundary value problems there are two different concepts for hierarchical type a posteriori error estimators due to Bank and Weiser [3] and Deuflhard, Leinen and Yserentant [9] Here, in case of mortar finite elements we first follow the ideas from [3] in so far as we consider a localization of the error equation on the subdomains level. Then, we proceed in much the same way as in [9] by approximating the individual error equations on the subdomains using higher order elements combined with a localization realized by means of a hierarchical ....

R.E.BANK AND A.WEISER; Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44, 283-301 (1985)

....by its elementwise contributions and provides lower and upper bounds for the discretization error. Here we use a hierarchical basis estimator based on the appropriate higher order space and restrict ourselves to the 2D case. We note that the concept of error estimator of this type is given in [9, 19] for conforming discretizations. It is further developed and analysed for more general discretization schemes like mixed or nonconforming methods in [10, 26, 27] We refer the reader to [41] for an overview and additional references. The starting point for hierarchical basis error estimators is ....

R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. (1985) 44, 283 -- 301.

....solution error e qn q in some norm. The choice of norm is restricted some what because the Helmholtz operator is not positive definite; thus an energy norm does not exist. Therefore a posteriori error estimators which measure error in an energy norm or otherwise assume positive definiteness [4, 5, 9, 11, 99] cannot be applied in a straightforward manner. Hsiao, et al. [67] derive a residual based a posteriori error estimator for the boundary integral equations, utilizing the natural norms of the operator and the solution. In particular, the residual measured in the H 1 2 norm approximates the error in ....

R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Mathematics of Computation, 44:283-301, 1985.

....survey on a posteriori error estimators see Verfurth [17] Several error estimators for boundary value problems have already been developed. For example, in the software package PLTMG [2] for solving elliptic partial differential equations, the triangle oriented error estimator of Bank and Weiser [4] is implemented, which is based on the solution of small Neumann problems on each triangle. In the finite element code KASKADE the local error estimator is edge oriented, see Deuflhard, Leinen and Yserentant [6] and Zienkiewicz et al. 20] For the generalized eigenvalue problem Friberg [8] ....

R.E. Bank, A. Weiser, "Some a--posteriori error estimators for elliptic partial differential equations", Math. Comput. 44,283 (1985).

....that the error estimates are given in terms of classical energy norm estimates of the errors in the numerical solution and numerical influence function. Reliable and accurate techniques have been developed to date to estimate the error in the global energy norm, using either residual methods (see [1 3,6,5,18]) or recovery methods (see [19,20] We also describe how to estimate lower and upper bounds of the error in the goal. A natural adjunct to this new error estimation approach is goal oriented adaptivity, where mesh adaptation is designed to accelerate the rate of convergence of the solution with ....

R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Math. Comp., 44:283--301, 1985.

.... same was done for high frequency scattering problems, i.e. the full Maxwell s equations, in [30] To the knowledge of the authors, hierarchical error estimator for quasistatic electromagnetics problems have not been investigated before, though they are widely used for ordinary elliptic problems [5, 21, 24, 32]. The paper is organized as follows: In the next section we introduce edge elements of lowest and second order. Then we discuss the principles of hierarchical error estima3 tion and present the concrete hierarchical error estimator. It turns out that it is essential to pay special attention to ....

R. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283--301.

....in Galerkin nite element methods is done with respect to the natural energy norm k k E induced by the underlying di erential operator, ku u h k E c S sup fk kE=1g jh (u h ) ij : 4.1. 3) This approach has been initiated by the landmark papers of Babu ska Rheinboldt [BR78] Bank Weiser [BW85], and Zienkiewicz Zhu [ZZ87] For discussions and more references see the survey articles by Ainsworth Oden [AO97] and Verf urth [Ver96] This approach seems rather generic as it is directly based on the variational formulation of the problem and allows to exploit its coercivity properties. ....

.... (3) K = c i h 1=2 K f K K gkn [rz (1) h ]k K ; where the cell residuals are de ned as above. In this case, we observe that lim inf TOL 0 I e 1 ( 1 8) 4. Approximation by local residual problems: Following the idea already used in the energy error estimator of Bank Weiser [BW85] (see also Ainsworth Oden [AO97] on each element K the local Neumann problems (rvK ; r h ) K = R(u h ) h ) K (r(u h ) h ) K 8 h 2 VK ; 4.2.28) are solved, where VK = fq 2 Q 2 (K) q Q 1 (K)g . The related error indicator based on solving patch wise Dirichlet problems is not ....

Bank R.E. and Weiser A. (1985) Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp. 44: 283-301.

....variational formulations (see [21] 22] and [25] for further details and applications) On the other hand, we also develop in this work an a posteriori error analysis for the nonlinear problem under consideration. We proved in [7] that one can combine the classical Bank Weiser method from [5] with the ideas from [11] and [13] to derive an a posteriori error estimate for the two fold saddle point formulation of an hyperelastic material in the plane. This analysis was recently extended in [24] and [6] to linear and nonlinear transmission problems in plane elastostatics. We prove in ....

....the method. 5 A posteriori error analysis In this section we follow the approach from [11] see also [7] 12] and [24] to derive a Bank Weiser s type a posteriori error estimate for the nite element solution introduced in Section 4. The original method due to Bank and Weiser was proposed in [5]. We begin with the following preliminary subsection. 13 5.1 Preliminaries The a posteriori error analysis to be developed here can be carried out for a general nonlinear operator N satisfying (2.1) 2.2) and other continuity assumptions. Nevertheless, for clarity of exposition we consider from ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations. Mathematics of Computation, 44 (1985), pp. 283-301.

.... for boundary integral equations, are given in [17] and [26] Also, explicit residual error indicators and hierarchical basis estimators for the combination of BEM with FEM and mixed FEM, are provided in [14] 15] 16] 24] and [27] More recently, the classical Bank Weiser estimator from [6] (see also [1] and [2] which involves the solution of equilibrated local Neumann problems, has been extended to a wider class of boundary value problems, including large strain elasticity, incompressibility, and interface problems (see, e.g. 9] 10] 11] and [22] In particular, we used ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations. Mathematics of Computation, 44 (1985), pp. 283-301.

....[1, 2] and the recent survey articles by Bornemann et al. 7] and Verfurth [32, 33] In this section we will focus on an element oriented error estimator which is based on the solution of local subproblems. In the standard conforming setting this kind of error estimator is due to Bank and Weiser [5]. It relies on a defect correction with respect to the higher order ansatz space of continuous, piecewise quadratics and an appropriate localization based on a hierarchical two level splitting. However, in contrast to the standard conforming setting we have to take into account the discontinuity ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)

....can be treated as in the conforming primal formulation whereas the second subproblem gives rise to an inde nite saddle point problem. 3 In sections 4 and 5, we present two types of hierarchical error estimators that are strongly related and require an adequate saturation assumption (cf. e.g. [6, 14, 15, 21, 25, 27]) In particular, for the derivation of the rst hierarchical error estimator which is dealt with in section 4 we start from an approximation of the defect problem in a higher order ansatz space followed by a localization in terms of an appropriate hierarchical two level splitting. For the ....

.... hier jjjj h h jjj div ) jjjj h jjj div : 2.11) Next, we consider an error estimator L based on the solution of local subproblems that is strongly related to the hierarchical error estimator. In the standard conforming setting, this kind of error estimator is due to Bank and Weiser [6]. For nonconforming techniques we refer to [21] It relies on 7 a defect correction with respect to a higher order ansatz space and an appropriate localization based on a hierarchical two level splitting. It turns out that the estimator can be computed elementwise by the solution of local ....

R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44 (1985), pp. 283-301.

....to a higher order finite element approximation combined with a suitable localization of that problem by a hierarchical two level splitting. In case of the standard conforming P1 approximation there are two different approaches: DDM on Nonmatching Grids 14 The first one due to Bank and Weiser [8] starts from a localization of the defect problem on the elements level resulting in local Neumann problems which are then solved by using continuous, piecewise quadratic elements and their standard hierarchical decomposition. The second technique which can be attributed to Deuflhard, Leinen, and ....

R.E.BANK AND A.WEISER; Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44, 283-301 (1985)

....In sections 3 and 4, we present two different hierarchical basis a posteriori error estimators. Both of them are based on a defect correction in a higher order space and a hierarchical twolevel splitting. This type of error estimator is well known for standard conforming discretizations [10, 17, 19, 20, 33]. An excellent overview of different techniques can be found in [33] see also the references therein) The starting point for the construction of the error estimators is the saddle Math. Institut, Universitat Augsburg, Universitatsstr. 14, D 86 159 Augsburg, Germany. E mail: ....

....the higher order case. 3. A Hierarchical Basis Error Estimator on Subdomains. In this section, we present a hierarchical basis error estimator which is based on a defect correction in an appropriate higher order space, a hierarchical splitting, as well as some localization techniques, cf. e.g. [9, 10, 19, 20, 33]. For standard conforming finite element discretizations there are basically two ways to obtain such an error estimator. One of them follows Bank and Weiser [10] where the defect problem is first localized and then discretized. Secondly, using the ideas of Deuflhard, Leinen, Yserentant [19] the ....

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R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985), pp. 283-301

.... and others (for an exhaustive bibliography see e.g. the recent monograph by Szab o and Babuska [11] For highly nonuniform triangular meshes, generated by the meanwhile standard refinement process of Bank and others [2] 3] appropriate error estimators have been developed by Bank and Weiser [4]. These error estimators which provide sharp lower and upper bounds for the global discretization error are element oriented in the sense that they are based on the elementwise solution of suitable low dimensional subproblems. An alternative approach developed by Deuflhard, Leinen and Yserentant ....

....mixed finite element techniques, since it is well known [1] 7] that by an appropriate post processing such methods are closely related to nonconforming discretizations. In particular, we will develop both element oriented and edge oriented error estimators similar to the approaches used in [4] and [9] In both cases the error equation will be approximated in the conforming finite element space of continuous, piecewise quadratics, since there is no canonical choice for a nonconforming piecewise quadratic ansatz. The paper is organized as follows: In section 2 we introduce the ....

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R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985) 26

....adaptive local grid refinement are well established tools in the efficient numerical solution of elliptic boundary value problems. In the framework of standard conforming finite element approaches we acknowledge the pioneering work due to Babuska and Rheinboldt [8, 9] and the more recent articles [11, 12, 35, 36, 54, 57]. Further references can be found in the survey article by Bornemann et al. 19] and in the excellent monography by Verfurth [55] In the context of nonconforming techniques we mention [41, 43] For mixed finite element methods involving Raviart Thomas elements we refer to [1, 25, 26, 42, 44, 45] ....

R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283--301.

....be neglected, if the triangulation is fine enough. We note that for each triangulation there is an example where this term is dominant and the error estimator fails. 6.2. Hierarchical error estimator. Hierarchical basis error estimators are well known for standard conforming discretizations [8, 40, 43, 90]. They are based on a defect correction in a higher order space and a hierarchical two level splitting. An excellent overview of different techniques can be found in [22, 90] see also the references therein) There are basically two ways to obtain such an error estimator. Using the ideas of ....

....in [22, 90] see also the references therein) There are basically two ways to obtain such an error estimator. Using the ideas of Deuflhard, Leinen, Yserentant [40] the resulting continuous defect problem is first discretized and then localized. The second possibility follows Bank and Weiser [8], where the defect problem is first localized and then discretized. These concepts have been generalized to nonconforming discretizations, 58, 95] mixed Raviart Thomas discretizations, 1, 29, 57, 60] and the Stokes problem [9, 88] In this section, we present a hierarchical basis error ....

[Article contains additional citation context not shown here]

R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283--301.

....will be developed by the concept of defect correction in higher order ansatz spaces and an appropriate localization by hierarchical two level splittings of these ansatz spaces. We remark that this concept is widely used in case of conforming or nonconforming finite element approximations (cf. e.g. [6], 7] 14] 19] 25] We assume that (T k ) l k=0 is a hierarchy of possibly highly nonuniform triangulations generated by the meanwhile standard refinement process due to Bank et al. 5] cf. also [4] 8] 14] 19] 28] In particular, a triangle K 2 T k , 0 k l, either remains ....

.... based on a reliable and efficient a posteriori error estimator for the total error in the flux which can be derived by the principle of defect correction in higher order ansatz spaces and an appropriate localization by hierarchical two level decompositions of these ansatz spaces (cf. e.g. [6], 7] 14] in case of conforming P1 approximations and [19] 26] for nonconforming P1 approximations) We denote by (j; u) 2 H(div ; Omega Gamma Theta L 2( Omega Gamma the unique solution of the mixed formulation (2.3) and by ( j 0 ; u 0 ) an iterative approximation of the lowest order ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)

.... Oswald s monograph [26] and the survey articles by Xu [32] Yserentant [34] and Zhang [36] Likewise, beginning with the pioneering work done by Babuska and Rheinboldt [2, 3] adaptive grid refinement based on appropriate a posteriori error estimators has attracted considerable interest (cf. e.g. [6], 8] 14] 24] 30] and the references therein) We note that the realization of adaptivity concepts and the multilevel iterative e mail: hoppe math.uni augsburg.de, wohlmuth math.uni augsburg.de address: Math. Nat. Fakultat der Universitat Augsburg, Universitatsstr. 14, 86 159 Augsburg, ....

....adaptive algorithm. We refer to the pioneering work done by Babuska and Rheinboldt [2, 3] and mention the recent survey article of Bornemann et al. 8] and Verfurth [29] In the case of standard conforming finite element discretizations different concepts have been established by Bank and Weiser [6], Deuflhard, Leinen, Yserentant [14] and Zienkiewicz and Zhu [37] However, in the context of mixed finite element techniques there is only some work of Braess and Verfurth [10] on residual based techniques, Braess et al. 9] and Maitre [25] In particular, we will focus in this section on the ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)

....started. During the last decades, fundamental approaches for analyzing a posteriori error estimators were developed, e.g. for residual based error estimators by Verf urth [17] and Johnson et al. 9] or for error estimators which are based on the solution of local problems by Bank and Weiser [4] and Ainsworth and Oden [1] These techniques can be applied to derive a posteriori error estimators in di erent norms for the Stokes equations. In this section we show that it is possible to develop a residual based a posteriori error estimator for jjg u g u h jj. Because the estimator in ....

R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial dierential equations. Math. Comp., 44:283-301, 1985.

....with various local error estimators and indicators, to develop adaptive nite element and nite volume codes for both 2 D and 3 D problems. The methods were based on the previous research by Babuska and Rheinboldt [4] 5] Zienkiewicz and Zhu [23] Becker and Rannacher [8] Bank and Weiser [7]; Brenner [10] We started the problem investigation by developing, implementing, and testing a 2 D grid re nement strategy based on the known error estimator and indicators. In the context of the nite element method, our work was mostly concentrated on the h version adaptive re nement, where ....

....reservoir with two wells. We extended the developed 2 D adaptive code to 3 D. We worked on, tested, and implemented three error indicators, namely residual based re nement ( rst introduced by Babuska and Rheinboldt [4] Zienkiewicz Zhu technique (see [23] and hierarchical re nement (see [6] [7]) We have adapted the rst two techniques for the nite volume element method. In R. Lazarov, S. Tomov, paper 21 of the List of Publications Resulting from This Grant) Adaptive nite volume element method for convection di usion reaction problems in 3 D, we present an adaptive numerical ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential problems, Math. Comp., 44 (1985), pp. 283-301.

....[1] 2] Recently, various posterior error estimators in connection with adaptive methods for mesh generation and mesh improvements have been proposed. Some of them are based on the continuity of the tangential component of the electric eld or the normal component of the magnetic ux density [3], and some are based on the uniformity of magnetic stored energy. However, two main points must be addressed: First, the error estimation must be independent of the type and class of problems solved and must operate uniformly over materials with various electromagnetic properties, and second, to ....

R. E. Bank and A. Weiser, \Some a posteriori error estimators for elliptic partial dierential equations,"Mathematics of computation, Vol. 44, No. 170, pp.283-301, 1985.

....65N12, 65N55 1. Introduction. The development of adaptive numerical methods is of enormous current interest. Although such concepts have not entered yet industrial applications at large, current research developments for instance in a nite element context indicate their very promising potential [1, 2, 7, 9, 47]. Such hopes and numerical experiences are, however, contrasted by negative statements proved in the context of complexity theory. In fact, on a rigorous level not much has been proved about adaptive nite element schemes in comparison with a priorily xed meshes. To our knowledge, the only result ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comput., 44 (1985), pp. 283-301.

....exhibit singularities. Much impetus to the theory of adaptive finite element methods was provided by the introduction and analysis of p and h p methods by Babuska and his collaborators (see e.g. 2] and [3] A lot of further results on this subject have been developed in the last years, see e.g. [4, 33, 52]. For an overview on the theory of adaptive finite elements, the reader is referred to [28] and [53] On the other hand, for most adaptive algorithms, there exist no proofs of convergence. The purpose of this paper is to phrase the problem of designing and analyzing adaptive methods in the ....

Bank, R. E. and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), 283--301.

.... depending on f and converging to 0 for h 0: For earlier work on a posteriori error estimators on isotropic meshes we refer to Babuska and Rheinboldt [2] and to the survey Verfurth [10] Of special interest are the residual based indicator of Verfurth [9] and the indicators of Bank and Weiser [4]. The crucial point of anisotropic a posteriori estimating is the fact that all classical estimators detoriate if the aspect ratio a( h 1 ( h 2 ( tends to infinity. Siebert [8] solves this problem by locally balancing the directional errors avoiding anisotropic overrefinement. On the other ....

....paper is as follows. In section 2, we show that the standard error estimator based on the residual does not satisfy (6) with constants m 1 ; m 2 independent of the aspect ratio a: Section 3 is devoted to the study of a nonlocal error estimator inspired by the third indicator of Bank and Weiser [4]. Despite the fact that the estimator is nonlocal, it is proved that it can be computed economically on isotropic and anisotropic meshes. On anisotropic meshes, the estimator shows a significant propagation of local errors along the small mesh direction e 2 ; which clearly indicates that local a ....

[Article contains additional citation context not shown here]

R. E. Bank and A. E. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, (1985), pp. 283--301.

....The equilibrium method for nding error estimators for the nite element method was introduced by P. Ladeveze in his doctoral thesis. A more accessible reference is [10] Later the approach was applied in a global setting by Kelly in for example [11] and in a local setting by Bank and Weiser in [12]. More recently the approach was applied again in a local setting by Ainsworth and Oden in [13] 14] and [15] and by Babu ska et al. in [6] 7] and [8] Most recently, the method has been treated at textbook level in [16] and [9] In all cases a very small class Date: August 18, 2000. 1 ....

....the minimization process. In the same paper it is suggested that in order to avoid solving a global problem, simple splitting functions like 1 2 can be used. Since we generally do not approve of global problems (and would like to do better than an even split) we turn to Bank and Weiser, who in [12], apparently independently of Kelly, present the same setting (74) but solve one by one the problems, Find ( 0) 2 U N e (J N ; I ) R k (v) S N k (v) S I k ( 0; v) 0; k j 1; j 2 s k ; 75) k = 1; N; for each pyramid function v 2 V N e . These ....

Randolph E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial dierential equations. Mathematics of Computation, 44(170):283-301, 1985.

....the leading order term is computable and has a proof of optimal order of accuracy, see Section 3. Error estimates based on solutions of local problems, i.e. local error indicators, have been widely analyzed and used for stationary elliptic problems provided with an energy norm, see [1] 2] 3] [4], 9] 12] 13] The computational domain is then subdivided and by providing approximate boundary conditions for the subdomains the local problems become decoupled. The error caused by the arti cial boundary condition can be of higher order than the approximation error in the interior. In this ....

R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp. 44 (1985), 283-301.

....jG respectively. Another proposed indicator hierarchically measures the L 2 error jH (E) kP V l 1v Gamma P V l vk L 2 (E) Related error criteria are for instance used in [19, 14] Hierarchical a posteriori error estimation measures the same difference, but mostly in a scaled H 1 norm [2]. We believe the first estimator to be the most reasonable for visualization purposes. Nevertheless, depending on the user s preferences, other indicators especially those mentioned here can be considered as well by our approach and implemented similarly. The results presented here are all ....

Bank, R. E.; Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44, 283-301, 1985.

....of the internal edges are connected by the de nition of X N;1 e (J N ; I ) Hence including the piecewise constant functions 1 0 k (de ned below (45) directly in ( Y N e ) i) leads to a global problem. The standard solution from the literature (see Kelly [20] Bank and Weiser [21], Ainsworth and Oden [22] 23] and [24] is to replace 1 0 k by the linear (for triangular elements) or bilinear (for rectangular elements) Lagrangian basis f kj g j2p k for k , extended by zero outside k to f 0 kj g j2p k . kj (P kl ) jl . We say that kj has Point of Attack in ....

Randolph E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial dierential equations. Mathematics of Computation, 44(170):283-301, 1985.

....refinement of the triangulations are an indispensable tool for efficient adaptive algorithms. Concerning the finite element solution of elliptic boundary value problems we mention the pioneering work done by Babuska and Rheinboldt [2, 3] which has been extended among others by Bank and Weiser [7] and Deuflhard, Leinen Yserentant [17] to derive element oriented and edge oriented local error estimators for standard conforming approximations. We remark that these concepts have been adapted to nonconforming discretizations by the authors in [29, 30] and [49] The basic idea is to discretize ....

R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)

....undertaken by Oswald [15] 16] and Zhang [20] Additionally we modify some ideas of Cowsar [8] which have been recently suggested in the framework of a domain decomposition approach. In the case of conforming methods an element oriented error estimator has been investigated by Bank and Weiser [3] while an edge oriented error estimator has been suggested by Deuflhard, Leinen and Yserentant [9] Both element and edge oriented error estimators provide sharp lower and upper bounds for the global discretization error. We have to modify the established techniques by taking into account the ....

....control the refinement process. A good error estimator should be easily computable and provide sharp lower and upper bounds for the energy norm. We recall that in the conforming case such error estimators are well established. An element oriented estimator has been investigated by Bank and Weiser [3] while an edge oriented estimator is due to Deuflhard, Leinen and Yserentant [9] 3.1 Element oriented error estimator We start with the construction of an element oriented estimator which follows the ideas of Bank and Weiser [3] We denote by uNC the available approximation of the nonconforming ....

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R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)

....tool for adaptive finite element codes, since its local contributions are used as indicators for local refinement of the triangulations. In the unconstrained case, i.e. for second order elliptic boundary value problems, element oriented and edge oriented error estimators have been proposed by Bank and Weiser (1985) and by Deuflhard, Leinen and Yserentant (1989) and have been implemented in the existing adaptive codes PLTMG and KASKADE, respectively (cf. also the KASKADE extension 3 D ELLKASK by Bornemann et al. 1993) in the 3 D case) Both estimators rely on a piecewise quadratic ansatz which is assumed to ....

Bank, R.E. and Weiser, H. (1985); Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301.

....error, typically in the energy norm, via a proper use of information extracted from both the computed solution and data. This can be phrased in terms of optimizing the computational effort for a given accuracy. One of the most successful estimators, implemented in [2] is due to Bank and Weiser [4]. It is obtained from the solution of local Neumann problems involving both a volumetric and jump (or line) residual of the finite element solution. It is inexpensive as compared with the solution process and performed elementwise, thereby leading to a computable estimate of the error. Even though ....

....and the true error is instead a more realistic property to aim at. It guarantees reliability and efficiency of associated mesh refinement algorithms [8] Equivalence for the Bank Weiser s estimator, hereafter called BW, has been derived in the energy norm under the so called saturation assumption [3,4]. This says that the solution can be approximated asymptotically better with quadratic than with linear finite elements. Although such a claim is true for functions smoother than those in H 2( Omega Gamma , it may not be valid for those intermediate between H 1( Omega Gamma and H ....

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R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 285--301.

....from both the computed solution and data. This can be rephrased in terms of optimizing the computational effort for a given accuracy, which in turn corresponds to avoiding overrefinement. Since the pioneering paper [3] a number of estimators have been proposed and tested for various PDEs [2,4,5,6,13,14,19,24,25]. Their success have originated an increasing interest in both applications of existing estimators and development of new ones, possibly for problems of different type or norms other than the energy norm. Pointwise error control, for instance, appears to be crucial for certain nonlinear problems ....

....between estimators and the true error is instead a more realistic property to aim at. It guarantees reliability and efficiency of associated mesh refinement algorithms [19] Equivalence has been derived for the energy norm under the sole assumption of mesh regularity in [2,22,24] and in [6] with an additional saturation assumption. In all these cases the estimators are computable 1991 Mathematics Subject Classification. 65N15,65N30,65N50,35B45. Key words and phrases. a posteriori error estimates, maximum norm, equivalence, point and line singularities. This work was partially ....

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R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), 285--301.

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R. E. Bank and A. Weiser, 1985. Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44 pp. 283-301.

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R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations, Math. Comp. 44 (1985), no. 170, 283--301.

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R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial di#erential equations. Math. Comp., 44(170):283--301, 1985.

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R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations, Math. Comp. 44 (1985), no. 170, 283--301. 0 20 40 60 80 100 120 140 160 180 200 220

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R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44:283-301, 1985.

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Bank, R.E., and Weiser, A. "Some A Posteriori Error Estimators for Elliptic Partial Di#erential Equations," Mathematics of Computations, Vol. 44, pp. 283301, 1985.

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Bank, R.E., and Weiser, A. "Some A Posteriori Error Estimators for Elliptic Partial Di#erential Equations," Mathematics of Computations, Vol. 44, pp. 283301, 1985.

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R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial di#erential equations, Mathematics of Computation, 44 (1985), pp. 283--301.

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Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations, Math. Comput., 44, 283--301 (1985)

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