By Weimin Han

This paintings presents a posteriori errors research for mathematical idealizations in modeling boundary price difficulties, specially these coming up in mechanical functions, and for numerical approximations of diverse nonlinear var- tional difficulties. An mistakes estimate is named a posteriori if the computed resolution is utilized in assessing its accuracy. A posteriori mistakes estimation is crucial to m- suring, controlling and minimizing blunders in modeling and numerical appr- imations. during this booklet, the most mathematical device for the advancements of a posteriori errors estimates is the duality idea of convex research, documented within the famous ebook by means of Ekeland and Temam ([49]). The duality thought has been stumbled on valuable in mathematical programming, mechanics, numerical research, and so on. The ebook is split into six chapters. the 1st bankruptcy reports a few simple notions and effects from sensible research, boundary price difficulties, elliptic variational inequalities, and finite aspect approximations. the main appropriate a part of the duality conception and convex research is in brief reviewed in bankruptcy 2.

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**Sample text**

If ai is not a node of K , then $ilK = 0. 30 We examine an example of linear elements. Assume R c IR2 is a poly onal domain, which is triangulated into triangles K , K E P h . e. e. the vertices of the triangulation that lie on 8 0 . From each vertex ai, construct K~ as the patch of the elements K which contain ai as a vertex. ~~ associated with the node ai is a continuous function in 2,which is linear on each K and . corresponding piecewise linear function space is is non-zero only in K ~ The then X h = span{41i, 1 5 i 5 Nh}.

We observe that if we keep the value of 11 u 11 L~ fixed, then the value 11 Qu I I L 2 ( n ) A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY will be minimal when the function u does not change rapidly. So we expect an optimal function u should keep its sign. 16). So the best constant is expected to be Let us show that these formulas indeed provide the best constant. , [61]). 18 I f v E H 1 ( f l ) ,then Ivl E ~ ' ( f l and ), i Vv Vlvl = 0 -Vv i f v > 0, i f v = 0, i f v < 0. From this lemma, we immediately obtain the next result.

A) Assume f (u)E R. Then f is continuous at u i f f is bounded from above in a neighborhood of u. (b) I f f ifJinite on an open set M C V and is continuous at some point of M , then f is continuous on M . 1 shows a convex function f that is continuous in the interior of its effective domain, and is not continuous at b ( f (x) = cc for x 2 b), a boundary point of the effective domain. Note that f is not bounded from above to the right of b. 18. 19 Let M C Rd be an open convex set. Then every convex function f : M -i R is continuous.