Zhou, G., Rannacher, R., Pointwise superconvergence of the streamline diffusion finite element method, Numer. Methods Partial Differential Equations (to appear). 28  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The fine part of the mesh is suitably aligned to give some resolution of the boundary layer, while outside that layer the mesh orientation is unimportant. The analysis of the streamline diffusion finite element method on coarse meshes that are aligned along the streamlines is considered in [12]. The piecewise uniform Shishkin mesh is specified a priori; thus, instead of pursuing a more complex adaptive mesh approach, we demonstrate in our analysis and numerical results that pointwise accurate numerical results can be obtained inside outflow boundary layers using simple meshes. 2 An ....

....that in this smooth region, the scheme is approximately second order as N 1, which is better than the order of 5=4 predicted by Theorem 5.2. A similar gap between theory and numerical experience is present in most other analyses of the streamline diffusion method (see, e.g. 5, 7] but see [12] for a special case and [11] for a full treatment of this question. We also tested the method on (6.5) and (6.6) with A = 0 on Omega Gamma In this case (6.6) gives a smooth exact solution to (6.5) The numerical results on Omega 0 for this smooth problem are identical to those displayed in ....

Zhou, G., Rannacher, R., Pointwise superconvergence of the streamline diffusion finite element method, Numer. Methods Partial Differential Equations (to appear). 28

....its corresponding one space dimension versions. Finite difference methods are discussed in [12, 16, 30] while finite volume method are discussed in [27] In the context of the finite element method, there is a variety of approaches, such as Petrov Galerkin FEM [18, 20] Streamline Diffusion FEM [21, 22, 46], High order FEM [1, 17] and Adaptive FEM [8, 11] to name but a few. For more details, see Hughes [19 ] Carey and Oden [6, Ch.5] and Zienkiewicz and Taylor [45, Ch.12] However few of the above methods are globally uniformly convergent (GUC) that is, the error between the original continuous ....

.... Hence to ensure global convergence, the mesh size h must be less than or equal to p ; where p is a positive number, which is impossible in practice, since usually can be as small as 10 Gamma10 : Hence much work switched to local error analysis, cf. Johnson et al. 22] Zhou and Rannacher [46] and for more details see Wahlbin [43] In the following, we will focus on GUC methods achieved by FEM. It is well known that uniform convergence can be achieved by using exponentially fitted splines or combinations with other functions as trial and test space [31, 34] However they are ....

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G. Zhou and R. Rannacher, Pointwise superconvergence of the streamline diffusion finite element method, Numer. Methods for PDEs 12(1996) 123-145. 37

....# respectively. These parts are to fulfill r k ch log 1 h and j r j c p h log 1 h . With these assumptions, the estimate ku Gamma u h k L 2( Omega 1 ) 6 Ch k 1=2 i kuk H k 1( Omega 2 ) kfk L 2 j holds. Rannacher and Zhou proved a similar estimate for the maximumnorm in [45]. It follows that influence of errors at point y on the solution at point x with jx Gamma yj = d decays exponentially with p d in crosswind and with d in upwind direction. This result corresponds to the observation, that information is transported only in streamline direction mimicking the ....

G. Zhou and R. Rannacher. Pointwise superconvergence of the streamline diffusion finite element method. Preprint 94--72, IWR, Universitat Heidelberg, December 1994.

....1 Gamma q i j i = 0; min (p; p 0 )g: 4.3) On such meshes, we will consider as trial spaces the space S p;1 0 (T ) cf. Fig. 1) We note that such mesh sequences would typically be generated by adaptive schemes that locate and try to resolve the layers. It can be shown using ideas of [7, 18, 19] (cf. 10] for the details) that the bubble stabilized hp SDFEM converges robustly and exponentially on compact subset of Omega for such mesh sequences: Theorem 4.1 Let a = 1, b = 0, q 2 (0; 1) 2 ( Gamma1; 1) be fixed. For p 2 lN consider the meshes T defined by the nodes (4.3) For a ....

Zhou, G., Rannacher, R., Pointwise Superconvergence of the Streamline Diffusion Finite Element Method,

....1 If T is not quasi uniform or fi is a variable, then let h be the diameter of local element and determine ffi s elementwise as in (6) see [14, p. 186] For this method with piecewise linear elements, pointwise error bounds of order O(h 2 j log hj) have been obtained for special meshes in [27], where it is also shown that the width of the characteristic boundary layers and interior layers along streamlines are of order h 5=8 log 2 h. See also [22, pp. 229ff. for discussion of such results. Assume Omega = 0; 1) Theta (0; 1) and a square mesh is generated by the nodes (x i ; y j ) ....

G. Zhou and R. Rannacher. Pointwise superconvergence of the streamline diffusion finite element method. Numer. Methods Partial Diff. Equations, 12:123--145, 1996.

.... 2 ; fi 1 ) is the crosswind vector and the coefficient of artificial crosswind diffusion is defined by m = for h 3=2 h 3=2 for h 3=2 : For this method with piecewise linear elements, pointwise error bounds of order O(h 2 j log hj) have been obtained for special meshes in [40], where it is also shown that the width of the characteristic boundary layers and interior layers along streamlines are of order O(h 5=8 log 2 h) See also [31, pp. 229ff. for discussion of such results. In our numerical experiments, we find that this method dramatically reduces the ....

....r j is evaluated within each element by setting ffi sc = fl 0 h maxf0; C 0 Gamma 1=P k g if ru h 6= 0 0 otherwise, 29) and C 0 = 0:7 for both linear and bilinear elements. For completeness, we show that SC CD discretization satisfies an error bound like those derived for SD in [1] 39] [40]. Assume that the quantity jfi r j satisfies q 0 jfi r j q 1 (30) for q 0 ; q 1 0, and for Gammar Delta fi d 0 0, let jjjvjjj 2 sc = krvk 2 ffi s kv fi k 2 q 0 ffi sc kv ff k 2 1 2 d 0 kvk 2 : 31) The following result establishes the stability of B sc defined by ....

G. Zhou and R. Rannacher. Pointwise superconvergence of the streamline diffusion finite element method. Numer. Methods Partial Diff. Equations, 12:123--145, 1996.

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