R. H. W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO, Modelisation Math. Anal. Numer. (1996) 30, 237 -- 263.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....some extra terms have to be added to the well known a posteriori error estimator used for the conforming case. In [11] 10] 8] these extra terms are the jumps across the element faces of the derivatives of the nite element solution in tangential direction with respect to the element faces. In [14], two other approaches for constructing an a posteriori error estimator are considered which are based on the solution of local subproblems or on a two level splitting of the space of piecewise quadratic conforming nite elements. For deriving an L 2 error estimator, John ( 15] 16] has used ....

R. H.W. Hoppe and B. Wohlmuth. Element-oriented and edge-oriented local error estimators for nonconforming nite element methods. RAIRO Model. Math. Anal. Numer., 30(2):237-263, 1996.

....this also gives an a posteriori error estimate for the discretization error u u (1) h of the nonconforming nite element method. This estimate can be regarded as an alternative to existing a posteriori error estimates for the nonconforming case which have been proposed for instance in [10] [11], 12] Moreover, the quantities post K can be used as local error indicators since they measure in some sense the local smoothness of the discrete solution u (1) h . The results concerning the a posteriori error control have been proved in a recent paper [17] The second application is the ....

....along the element face E. For this estimator, it holds ju u h j 1; C 7 conf : 50) However, if u N h 2 V (1) h 6 H 1 0( is a nonconforming nite element approximation of the weak solution u 2 H 1 0( of problem (47) then it is also well known from the literature (see [10] 9] [11]) that the estimators conf K have to be modi ed into e K by adding some extra terms containing the jumps of the derivatives of u N h in the tangential directions t E of the element faces E 2 E(K) For the corresponding modi ed estimator e : P K2T h e 2 K 1=2 the a posteriori ....

R. H.W. Hoppe and B. Wohlmuth. Element-oriented and edge-oriented local error estimators for nonconforming nite element methods. RAIRO Model. Math. Anal. Numer., 30(2):237-263, 1996.

....some extra terms have to be added to the well known a posteriori error estimator used for the conforming case. In [11] 10] 7] these extra terms are the jumps across the element faces of the derivatives of the nite element solution in tangential direction with respect to the element faces. In [15], two other approaches for constructing an a posteriori error estimator are considered which are based on the solution of local subproblems or on a two level splitting of the space of piecewise quadratic conforming nite elements. For deriving an L 2 error estimator, John ( 16] 17] has used ....

R. H.W. Hoppe and B. Wohlmuth. Element-oriented and edge-oriented local error estimators for nonconforming nite element methods. RAIRO Model. Math. Anal. Numer., 30(2):237-263, 1996.

....with respect to h, sup 2W N h P K2T h (div ; q) K k k W N h fikqk; q 2 Q h : 1.5) While a priori theory for these element pairs, stability and error analysis, is known, there are still only few a posteriori error estimates. Estimates exploiting local auxiliary problems are presented in [5, 6] for triangular elements. We take a different approach and derive weighted residual type estimates as described in [1, 2, 7, 10] Since there is no Galerkin orthogonality, that is, a h ( Phi u Gamma u N h ; p Gamma p h Psi ; f h ; q h g) 6= 0 (1.6) for arbitrary functions h 2 W N h , ....

R. H. W. Hoppe and B. Wohlmuth. Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO, M 2 AN, 30(2):237--263, 1996.

....progress has been made in the past twenty years. Emphasis on a posteriori error estimators such that do provide both local and global information on the error of the numerical solution and can be cheaply computed by the available numerical approximation and the data of the problem (cf. e.g. [5 10, 19, 23, 26, 27, 34, 41]) The most common error estimators are residual based estimators arising from an appropriate evaluation of the residual of the computed approximation, hierarchical estimators that can be derived by approximating the error equation, using higher order finite element spaces combined with suitable ....

....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 generally a saturation assumption. However it can often be shown that the contributions are locally equivalent to those of a residual based a posteriori error ....

R. H. W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO, Modelisation Math. Anal. Numer. (1996) 30, 237 -- 263.

....survey article [39] We refer to the authors article [35] for a comparison of these preconditioners. However, for nonconforming discretizations less work has been done concerning the realization of adaptive grid refinement based on appropriate a posteriori error estimators. In particular, in [18] the authors have developed both edge oriented and element oriented a posteriori error estimators using the well known principle of defect correction in a higher order ansatz space adapted to the nonconforming setting. In this paper, we shall be concerned with the construction and implementation ....

....Germany. E mail: hoppe math.uni augsburg.de, wohlmuth math.uni augsburg.de similar ideas cf. also Sarkis [29] The multilevel iterative solution process will go hand in hand with an adaptive grid refinement technique relying on the element oriented error estimator developed by the authors in [18]. We consider the following boundary value problem for a linear second order elliptic differential operator Lu : Gammadiv (aru) bu = f in Omega ; u = 0 on Gamma : Omega (1.1) where Omega stands for a bounded, polygonal domain in the Euclidean space IR 2 and f 2 L 2 ( Omega Gamma7 ....

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R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. To appear in M 2 AN Math. Modelling and Numer. Anal.

....around. It uses a conforming P2 approximation of the global defect problem followed by a localization of the discretized problem by means of the hierarchical two level splitting. Note that hierarchical type error estimators for nonconforming finite element discretizations have been considered in [22, 23]. Here, in case of the mortar finite element approximation we suggest a somewhat hybrid approach in so far as we begin with a localization of the defect problem (4.3) on the subdomain level and then consider a conforming P2 approximation of the resulting Neumann problems on the individual ....

R.H.W.HOPPE AND B.WOHLMUTH; Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. M 2 AN Math. Modeling and Numer. Anal. 30, 237-263 (1996)

....Weiser [10] where the defect problem is first localized and then discretized. Secondly, using the ideas of Deuflhard, Leinen, Yserentant [19] the resulting continuous defect problem is first discretized and then localized. These concepts have been generalized for nonconforming Crouzeix Raviart, [24], and mixed Raviart Thomas discretizations, 2, 25, 35] Here, we will use a combination of both techniques. In a first step, the continuous defect problem will be localized for each subdomain Omega i , 1 i K. We then use a higher order finite element discretization as well as an approximation ....

R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO, Modelisation Math. Anal. Numer. 30 (1996), pp. 237-263

....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 unrefined or is ....

.... 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 unrefined or is subdivided into four congruent subtriangles (regular or red refinement) or is bisected into two subtriangles (irregular or green refinement) The subtriangles are referred to as regular or irregular triangles, ....

[Article contains additional citation context not shown here]

R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite elements methods. Submitted to M 2 AN Math. Modelling and Numer. Anal.

....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 the defect problem for the available approximation with respect to a finite element space of higher accuracy. For a detailed representation of the 15 different concepts and further references we refer to the monographs of Johnson [31] Szabo and Babuska ....

R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite elements methods. Submitted to M 2 AN Math. Modelling and Numer. Anal.

....holds cjku Gamma uNC jk jke 2 Q jk jku NC Gamma u h jk (3.7) jke 2 Q jk c(jku Gamma uNC jk jku NC Gamma u h jk) 3.8) where c and c are positive constants depending only on the shape regularity of 2 T 0 and on the ellipticity of the problem but not on the refinement level. See [12] for the assumptions and a detailed proof. e 2 Q is only of academic interest and has no practical importance, because the computation of e 1 Q requires the solution of a 6 Theta 6 linear system on each inner element. We can reduce the computational amount by means of the elliptic projection ....

....Q Gamma uNC jk 2 r T H Gamma1 BPX 0 0 D Gamma1 QQ r jku NC Gamma QEuNC jk 2 ; 3.11) where HBPX stands for the conforming BPX preconditioner matrix and DQQ for the diagonal part of the quadratic surplus stiffness matrix. For a detailed specification of Q P and QE see [12]. Altogether we have developed an appropriate two sided error estimator which can be calculated edgewise if the iteration error is small enough. 4 Numerical Results We use the following test problem to illustrate the performance of the preconditioned iterative solvers and the reliability of the ....

R.H.W. Hoppe and B. Wohlmuth, Element-oriented and Edge-oriented Local Error Estimators for Nonconforming Finite Element Methods. TUM-M9206 Technische Universitat Munchen (1992), submitted to M 2 AN

.... concepts as developed by Bank and Weiser [5] and by Deuflhard, Leinen, and Yserentant [15] We further remark that hierarchical type a posteriori error estimators for standard nonconforming finite element discretizations of elliptic boundary value problems have been established by the authors in [19, 20]. We assume that (T (k) i ) k2N0 are regular, locally quasiuniform, nested sequences of simplicial triangulations of Omega i ; 1 i N . We denote by E (D) k the sets of edges of T k = N i=1 T (k) i in D Omega and by 0 c C generic constants that only depend on the shape regularity ....

R.H.W.HOPPE AND B.WOHLMUTH; Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. M 2 AN Math. Modeling and Numer. Anal. 30, 237-263 (1996)

....Weiser [10] where the defect problem is first localized and then discretized. Secondly, using the ideas of Deuflhard, Leinen, Yserentant [19] the resulting continuous defect problem is first discretized and then localized. These concepts have been generalized for nonconforming Crouzeix Raviart, [23], and mixed Raviart Thomas discretizations, 2, 24, 32] Here, we will use a combination of both techniques. In a first step, the continuous defect problem will be localized for each subdomain Omega i , 1 i K. We then use a higher order finite element discretization as well as an approximation ....

R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO, Modelisation Math. Anal. Numer. 30 (1996), pp. 237-263

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