I. Babuska and W.C. Rheinboldt. Error estimates for adaptive finite element computation. SIAM Journal of Numerical Analysis, 15, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....certain more theoretically based estimators can alternatively be explained as smoothening estimators. A second approach to the computation of error estimators is taken in the residual methods where the solution of certain local problems on each element gives the error estimation (see for example [6], 5] 3] and [2] The literature is immense and the references cited only provide a few examples. Common to all of the above mentioned approaches is that they provide estimation of the error in one particular norm (normally the energy norm) Change of norm is possible only with (often ....

Ivo Babuska and Werner C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM Journal of Numerical Analysis 15 (1978), no. 4, 736--754.

....Gradient recovery, ZZ patch recovery, a posteriori error estimate, finite element method AMS Subject Classification. 65N30, 65N15 1. Introduction. Since the publication of the pioneer work on a posteriori error estimates and adaptive finite element methods in 1978 by Babuska and Rheinboldt [4], more than two decades has past. The field now is in its maturity. Many methods have been developed including explicit error estimators, residual type error estimators, recovery type estimators, and error estimators based on hierarchic basis. For the literature, readers are referred to recent ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no.4, 736754.

....general result is provided in [19] to tackle the additional geometric truncation error. On the other hand, the constant C is not quantified in Theorem IV.1. In the FEM literature, more precise and specific estimation methods do exist to accurately quantify the discretization error (for instance, [20], 21] In this section, our goal is slightly di#erent: it is rather to propose qualitative and quantitative simulation results that illustrate the convergence of a FEM direct model in the context of EIT. A. Discretization in the case of an explicit solution A very simple way to estimate the ....

I. Babuska and W. Rheinboldt, "Error estimates for adaptive finite element computations", SIAM Journal of Numerical Analysis, vol. 15, no. 4, pp. 736--754, August 1978.

....error estimates are computable quantities that measure the actual errors without knowledge of the limit solution. They are essential in designing algorithms for mesh and time step modification which equidistribute the computational effort and so optimize the computations. Since the seminal paper [2] on elliptic problems there has been an ever increasing interest on the development of reliable and efficient adaptive algorithms for various linear and nonlinear PDEs. In particular, a posteriori error estimates have been derived in [14] 15] for linear and mildly nonlinear parabolic problems, ....

I. Babuska and C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM

....estimates are computable quantities that measure the actual errors without knowledge of the limit solution. They are instrumental in devising algorithms for mesh and timestep modification which equidistribute the computational effort and so optimize the computations. Ever since the seminal paper [1] on elliptic problems, adaptivity has become a central theme in scientific and engineering computations. In particular, a posteriori error estimators have been derived in [8] 9] for linear and mildly nonlinear parabolic problems, and in [20] 3] for degenerate parabolic problems of Stefan type. ....

I. Babuska and C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1987), pp.736-754.

....An important point to control the quality of the error estimator is to consider 1 Supported by the Swiss National Science Foundation. Preprint submitted to Elsevier Preprint 4 October 2001 the e#ectivity index defined by e# = estimated error true error . This subject has been initiated in [1,2] and extended to linear elliptic problems, see for instance [3 8] and to nonlinear elliptic problems [9 12] Several a posteriori error estimates have been derived for parabolic problems. In [13,14] a posteriori error estimates are derived for linear and nonlinear parabolic problems when using ....

....(0,1) and that # 0 , c 0 are between 0 and 1, 12) so that, according to [23] theorem 3) a maximum principle holds and #, c are also between 0 and 1. Thus, the solution to problem (1) 4) has a physical meaning and all the nonlinear terms in (7) can be truncated to zero outside the interval [0, 1]. This allows assumptions (10) and (12) to be fulfilled, and the existence result to hold. Finally, this being useless in our proofs, we mention that the space regularity of c is one order lower than the space regularity of #, due to the div (D 2 (c, #)##) term in eq. 2) Let us now turn to the ....

I. Babuska and W.C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15:736--754, 1978.

....solution algorithms, another type of error estimates, called a posteriori error estimate, are needed. An a posteriori error estimate can be computed once the numerical solution is known. The pioneering work on a posteriori error analysis for Finite Element Method was done in late seventies (see [2, 3]) Summarizing accounts of the topic can be found in [1, 7, 15] In [12] a posteriori error estimates with local weights are derived to adaptively solve problems in perfect plasticity. The organization of the paper is the following. In the next section, we will review the plasticity problem of ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 18 (1978), 736--754.

....CR BPX and Gamma HB : Gamma S HB , Gamma BPX : Gamma S BPX 2 3. ERROR ESTIMATOR BASED ON LOCAL SUBPROBLEMS Reliable and efficient a posteriori error estimators are an indispensable tool for efficient adaptive algorithms. We refer to the pioneering work done by Babuska and Rheinboldt [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 ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978)

....the jjj : jjj norm. The first estimator relies on a proper evaluation of the residual with respect to the mortar finite element approximation. We note that the concept of residual based error estimation DDM on Nonmatching Grids 10 can be traced back to the early work by Babuska and Rheinboldt [5, 6] and has been subsequently further developed and analyzed by various authors [9, 21, 36] In case of nonconforming finite element methods, such estimators have been recently considered in [17] For a comprehensive treatment and additional references we refer to [37] We assume that (T (k) i ) ....

I.BABUSKA AND W.RHEINBOLDT; Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978)

....estimators. We remark that such techniques are well developed for the standard conforming finite element methods. In particular, we refer to the recent survey articles of Xu [27] Yserentant [29] and Zhang [30] concerning multilevel methods and to the pioneering work of Babuska and Rheinboldt [2], 3] and the recent papers of Bornemann et al. 7] and Verfurth [25] for the issue of error estimation. However, less work has been done in the framework of mixed methods. We mention a multigrid approach proposed by Brenner [10] and additive as well as multiplicative Schwarz iterations developed ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978)

....by standard conforming finite element methods has been extensively dealt with. For an overview and further references we refer to 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: ....

.... P l k=1 e j k satisfies (3.6) 17 4 A Posteriori Error Estimator The adaptive refinement process is controlled by an a posteriori error estimator. Local refinement is an indispensable tool for an efficient 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 ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978)

....on a rigorous level and thereby complement the intruiguing adaptive algorithmic developments discussed before. On the other hand, adaptive techniques have been extensively studied in the context of finite element discretizations of (primarily) elliptic differential equations; see, for instance, [8, 9, 11, 24, 91, 178]. These methods are based on a posteriori error indicators or estimators. In practice they have been proven to be quite successful. However, the analysis and the schemes are rather dependent on the particular problem at hand and on the particular type of finite element discretization. The ....

I. Babuska, W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736--754.

....experiments. Numerical experiments indicate that the method is valid beyond the limits of the theoretical analysis, justifying this heuristic approach. The limits of validity of this method are a continuing topic of investigation. This circle of ideas was pioneered by Babuska and co workers [32, 33, 34]. An application to compressible fluid flow is given in [35] The Burgers equation is studied in [36] See also [37] To illustrate the idea of a posteriori estimates, we formulate them for the nonlinear conservation law U t rF (U) 0 : Although a posteriori estimates have not been studied ....

I. Babuska and W. Rheinboldt. Error estimates in adaptive finite element computations. SIAM J. Numer. Anal., 15:736--754, 1978.

....eight subvertices are its offspring. The eight vertices having a common parent are called siblings. The 64 vertices having a common grandparent are called cousins. A grid is said to be uniform if all its elements are at the same level. A grid is admissible in the sense of Babuska and Rheinboldt [3, 4] if it is defined recursively by following two rules: 1. Omega is an admissible grid; 2. If Delta Omega is an admissible grid and Delta is an element of Delta Omega then the grid obtained from Delta Omega and the eight elements created by trisecting Delta is admissible. Such grids contain ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736-754.

....of both. Two vertices are said to be edge neighbors if they are at the same level in the tree and if their intersection contains a edge (but not a face) of both. A grid is said to be uniform if all its elements are at the same level. A grid is admissible in the sense of Babuska and Rheinboldt [3, 4] if it is defined recursively by following two rules: 1. Omega is an admissible grid; 2. If Delta Omega is an admissible grid and Delta is an element of Delta Omega then the grid obtained from Delta Omega and the eight elements created by trisecting Delta is admissible. Such grids contain ....

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736-754.

....briefly introduce the discretization of the Stokes equations by stabilized finite elements. The following section presents the estimator for the discretization error. It is a generalization of the well known residual based indicator for the Poisson equation, as proposed by Babuska and Rheinboldt [4] or Eriksson and Johnson [22] For the Stokes equations, a posteriori estimates have been presented by Verfurth [56] and Bernardi et al. 5] Usually the velocity error is estimated in the H 1 and the pressure error in the L 2 norm. As the quantity of physical interest in flow computations ....

Babuska, I., Rheinboldt, W.C. : Error estimates for adaptive finite element computations, SIAM J. Num. Anal. 15, 736-754 (1978).

....threshold. A simple, but usually quite satisfying choice is to set j to a fixed percentage of the maximum error encountered on the mesh: j = 1 4 max T T (11) The factor 1 4 is empirical. An attractive alternative is to use extrapolated local errors for the determination of the limit [BR78]. For the element error T we assume a local behaviour of the form T = c T h P T T If element T was created by a subdivision of its father T 0 , we can predict a value t for the sons of element T t = 2 T T 0 Then we set the limit j to j = 1 2 max t t (12) The factor ....

I. Babuska and W. D. Rheinboldt. Error estimates for adaptive finite element computation. SIAM J. Numer. Anal., 15, 1978.

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I. Babuska and W.C. Rheinboldt. Error estimates for adaptive finite element computation. SIAM Journal of Numerical Analysis, 15, 1978.

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I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736--754.

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I. Babuska and W. C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15:736--754, 1978.

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I. Babuska and W.C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15:736--754, 1978.

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I. Babuska and W. C. Rheinboldt, "Error Estimates for Adaptive Finite Element Computations," SIAM J. Numer. Anal. 15, 736--754 (1978).

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I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736--754.

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Babuska, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations, SIAM J. Num. Anal., 15, 736--754 (1978)

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I. Babuska and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), pp. 736-754.

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