By Repin, Sergey

This ebook offers with the trustworthy verification of the accuracy of approximate options that's one of many vital difficulties in sleek utilized analysis. After giving an summary of the equipment built for types in response to partial differential equations, the writer derives computable a posteriori errors estimates through the use of equipment of the idea of partial differential equations and useful research. those estimates are appropriate to approximate suggestions computed through numerous tools.

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**Sample text**

The functional J W V ! 38) v2V is said to be dual (or conjugate) to J . 1. If J is a smooth function that increases at infinity faster than any linear function, then J is the Legendre transform of J . , see [121, 132, 324]). The functional J is also called polar to J . v /g v 2V is called the second conjugate to J (or bipolar). If J is a convex functional attaining finite values, then J coincides with J . To illustrate the definitions of conjugate functionals, consider functionals defined on the Euclidean space E d .

Babuˇska and T. Strouboulis [35]. 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 [338], R. Verf¨urth [358, 360], and M. Ainsworth, J. T. Oden and C. Y. Lee [9]. 3 A posteriori estimates based on post-processing of approximate solutions Post-processing methods exploit certain a priori known properties of exact solutions. In general terms, the situation that typically arises for finite element approximations is as follows.

36) Henceforth, we assume that the supremum (or infimum) of a quotient is taken with respect to all elements of V , except for the zero element 0V . Any affine functional defined on elements of V has the form hv ; vi ˛, where v 2 V and ˛ 2 R. , if there exists a one-to-one mapping of V to V and back that preserves the metric). All Hilbert spaces are reflexive. The same is true for the spaces Lp with 1 < p < C1. The theorem of F. 37) where u is uniquely determined. The functional J W V ! 38) v2V is said to be dual (or conjugate) to J .