By Weimin Han

ISBN-10: 144193636X

ISBN-13: 9781441936363

This quantity presents a posteriori mistakes research for mathematical idealizations in modeling boundary worth difficulties, in particular these bobbing up in mechanical functions, and for numerical approximations of various nonlinear variational difficulties. the writer avoids giving the implications within the so much common, summary shape in order that it really is more uncomplicated for the reader to appreciate extra basically the fundamental rules concerned. Many examples are integrated to teach the usefulness of the derived mistakes estimates.

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In the special case when the bilinear form a ( . 48) Preliminaries where the "energy functional" is defined as Now with a finite dimensional subspace VN c V chosen, it is equally natural to develop a numerical method by minimizing the energy functional over the finite dimensional space VN, UN E V N , E ( u N ) = inf E ( v ) . 50) are equivalent. The method based on minimizing the energy functional over finite dimensional subspaces is called the Ritz method. From the above discussion, we see that the Galerkin method is more general than the Ritz method, while when both methods are applicable, they are equivalent.

41) has a solution u which is sufficiently smooth so that all the following calculations are valid. 33), + jflC E( u ). E ( V- u ) dx. Now consider the integral on r c . 41), we have u , . u , = -gIuTl o n r c . This can be deduced by distinguishing two cases: la, / Hence, using the decomposition < g and lurl = g. 45). 41) reads: Find u E V such that Due to the assumptions on C and meas(rD) > 0, the bilinear continuous form a ( . 42). It can be easily checked that t : V -+ R is a linear continuous functional and j : V -+ is a proper lower semicontinuous convex functional.

S. ) i f for any sequence {u,) c V with u, u in V , f( U) < lim inf f - n-+m (u,). EXAMPLE2 . 9 Let K C V. , whereas K is a convex set iff IK is a convex function. e. the set of the points in K such that each point is contained in an open ball that in turn lies in K. Roughly speaking, int K is the subset of K excluding the boundary points. Some of the boundary points may not belong to K , unless K is a closed set. , the union of the set K and its boundary. The following results are not difficult to prove.

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A Posteriori Error Analysis via Duality Theory by Weimin Han

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