By Repin, Sergey
This e-book offers with the trustworthy verification of the accuracy of approximate strategies that's one of many critical difficulties in glossy utilized research. After giving an outline of the tools constructed for types according to partial differential equations, the writer derives computable a posteriori blunders estimates by utilizing equipment of the idea of partial differential equations and practical research. those estimates are acceptable to approximate options computed via numerous methods. Read more...
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Additional info for A posteriori estimates for partial differential equations
2) have an essential drawback: they are valid only for q 2 Qf . However, the set Qf is defined by the differential relation, which in general is difficult to exactly satisfy. In S. Mikhlin , a strategy close to the known orthogonal projections method (see also M. Vishik , H. Weil , S. Zaremba ) is discussed. It is based on the construction of an approximating sequence for the dual problem. However, practical realization of this approach within the framework of locally supported finite element approximations may be faced with serious technical difficulties.
Rd / (globally or locally). Another option is to set U D Qf . These observations suggest an idea to post-process ruh and find a close vector-valued function that satisfies some of the above-menioned properties. Formally, the principal scheme is as follows. ƒvh / is much closer to ƒu than ƒvh . ƒvh / cator of element-wise errors. ƒvh generates an efficient indi- Regularization. , see the paper by J. H. Bramble and A. H. Schatz , which is one of the earliest publications in this area). If the error caused by violations of a priori regularity properties dominates and a postprocessing operator efficiently performs regularization of approximate solutions, then one may hope that the difference between the approximate solution and its regularized (smoothed) counterpart represents the major part of the error.
Finally, we note that implicit type methods are often used for the indication of local errors. Concerning a posteriori methods developed to evaluate local errors of FEM approximations, we address the reader to the books by M. Ainsworth and T. Oden  and I. Babuˇska and T. Strouboulis . Also we recommend papers by I. Babuˇska, F. Ihlenburg, A. Mathur, T. Strouboulis, S. K. Gangaraj, C. S. Upadhyay [27, 39, 39, 37], E. Stein and S. Ohnimus , R. Verf¨urth [358, 360], and M. Ainsworth, J.
A posteriori estimates for partial differential equations by Repin, Sergey