M. Dobrowolski, S. Graf and C. Pflaum, A Posteriori error estimators in the finite element method on anisotropic meshes, ETNA 8 (1999) pp. 36-45.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by a factor which is (at most) proportional to the aspect ratio of the anisotropic elements. This potentially unbounded factor renders the isotropic, conventional estimators useless. Hence in the last decade there has been increasing research to find adapted estimators for anisotropic meshes, cf. [Sie96, Kun99, Kun00, Kun01, FPZ01, DGP99, Ran01]. It turns out that the upper error bound is the crucial issue which involves a proper alignment of the anisotropic mesh, see e.g. Kun00] Thus we may examine the existing approaches to derive upper error bounds for the Stokes problem on isotropic meshes. One encounters just a few techniques ....

M. Dobrowolski, S. Graf, and C. Pflaum. On a posteriori error estimators in the finite element method on anisotropic meshes. Electronic Transactions Num. Anal., 8:36--45, 1999.

....of APEE s. For isotropic grids, many different APEE s have been already proposed for the Stokes problem, see [Ver89] Ver91] Ain97] JL00] and many others. In the context of anisotropic meshes, there are already a variety of APEE s for Poisson and reaction diffusion problems ( Kun97] KV00] [DGP99], Kun00] Kun01] But for the Stokes equation, APEE s have not yet been inspected in anisotropic meshes. In section 2, we recall the definition of an anisotropic mesh and we introduce various definitions and notations. Section 3 treats exclusively the strengthened Cauchy Schwarz inequality for ....

M. Dobrowolski, S. Graf, and C. Pflaum. A posteriori error estimators in the finite element method on anisotropic meshes. ETNA, 8:36--45, 1999.

....sense, the criterion used is only indirectly related to the error. Recent work in error estimation is starting to reveal the explicit dependence of the error on both solution derivatives and on the mesh. This work shows great promise but is still in its early stages. Apel, Kunert and Dobrowolski [1, 13] have developed promising estimates for Poisson type problems which explicitly indicate mesh and solution dependence. Bank and Smith [3] in error analysis for the method used in the PLTMG code, show how the error can be written using the second derivatives of the solution along edges, d i , and ....

M.Dobrowolski, S.Graf and C.Pflaum, On a Posteriori Error Estimators in the Finite Element Method on Anisotropic Meshes. Electronic Transactions on Numerical Analysis, Vol. 8, 36-45, 1999. 20

....years. So far, many applications of anisotropic elements utilize heuristic arguments and lack rigorously analyzed error estimators. Some strictly mathematically based estimators have appeared recently however, due to Siebert [16] Kunert [13, 14] Kunert and Verfurth [15] and Dobrowolski et al. [9]. Before discussing these estimators in more detail, let us comment on an important feature that seems to be inherent in anisotropic error estimators. The theory of these estimators is not as complete as for isotropic elements since, at this time, no error estimator is known that bounds the error ....

....and analyzes an L 2 residual error estimator for tetrahedral meshes that bounds the error (in the L 2 norm) from above and below. Additionally, the face residuals (i.e. the gradient jumps) alone suffice to define this estimator, see Kunert and Verfurth [15] for further details. Dobrowolski et al. [9] also investigate the Poisson problem on triangular meshes. Applying the methodology of Bank and Weiser, they derive a global error estimator by solving a global problem. The (global) error bound relies on a saturation assumption that again requires a suitable anisotropic mesh. Note that no local ....

M. Dobrowolski, S. Graf and C. Pflaum (1999), On a posteriori error estimators in the finite element method on anisotropic meshes. Elec. Trans. Numer. Anal., 8:36--45.

....evident. The reason is that the common error estimators for isotropic meshes fail, or their proofs are no longer valid since the aspect ratio can be unbounded. We may stress here that very few a posteriori error estimators for anisotropic meshes are known that are rigorously analysed and proven [Sie96, Kun99a, Kun99b, KV99, DGP99]. In other papers more or less convincing heuristic arguments are given. Considering the whole adaptive algorithm, it is a quite natural desire that the error estimation (step 2) should provide all necessary information for step 3, in particular the (quasi) optimal element size, the stretching ....

....derivative) shows little change. Mathematically, Siebert [Sie96] restricts the set of treatable anisotropic functions, and Kunert [Kun99a, Kun99b] introduces the matching function m 1 (v; T ) that measures the correspondence between an anisotropic function v and an anisotropic mesh T . Lastly, in [DGP99] a saturation assumption is necessary that implies a similar correspondence. Despite these different descriptions, the known results strongly indicate that an anisotropic mesh has to correspond to the anisotropic function in order to obtain reliable and efficient error bounds. In the context of ....

M. Dobrowolski, S. Graf, and C. Pflaum. On a posteriori error estimators in the finite element method on anisotropic meshes. Electronic Transactions Num. Anal., 8:36--45, 1999.

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M. Dobrowolski, S. Graf and C. Pflaum, A Posteriori error estimators in the finite element method on anisotropic meshes, ETNA 8 (1999) pp. 36-45.

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