A Posteriori Estimates for Partial Differential Equations - download pdf or read online

By Repin, Sergey

ISBN-10: 3110191539

ISBN-13: 9783110191530

This ebook offers with the trustworthy verification of the accuracy of approximate ideas that's one of many valuable difficulties in smooth utilized analysis. After giving an summary of the equipment constructed for types in response to partial differential equations, the writer derives computable a posteriori mistakes estimates by utilizing tools of the speculation of partial differential equations and useful research. those estimates are appropriate to approximate strategies computed by means of quite a few equipment.

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Extra resources for A Posteriori Estimates for Partial Differential Equations (Radon Series on Computational and Applied Mathematics)

Sample text

It is easy to see that this functional is equal to zero if v coincides with u. 3) j Fv j WD sup w2V0 ; krwk w6D0 is positive. Therefore, it is natural to call Fv the error functional. It is easy to show that j Fv j is indeed a measure of the deviation of v from u. 4). Thus, the norm of the norm of the deviation from the exact solution coincides with the norm of Fv . Here arises the problem of how to compute j Fv j practically for a given v. 3) is hardly possible. A more promising way is to find computable upper bounds of j Fv j .

By definition, the space V consists of all linear continuous functionals on V . It is called topologically dual to V . The value of v 2 V on v 2 V is denoted by hv ; vi. This product generates a duality pairing of the spaces V and V . 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 . 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 .

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A Posteriori Estimates for Partial Differential Equations (Radon Series on Computational and Applied Mathematics) by Repin, Sergey

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