By Repin, Sergey

This e-book bargains with the trustworthy verification of the accuracy of approximate suggestions that is one of many imperative difficulties in smooth utilized research. After giving an summary of the equipment constructed for versions in keeping with partial differential equations, the writer derives computable a posteriori errors estimates by utilizing equipment of the speculation of partial differential equations and practical research. those estimatesRead more...

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**Extra info for A posteriori estimates for partial differential equations**

**Example text**

See C. Johnson and B. Mercier [188]) are constructed by means of macroelements. Sometimes, numerical analysts use “almost equilibrated” approximations that may not exactly satisfy the differential relation. 2) is not guaranteed. Also, it is worth remarking that an upper bound constructed by some q 2 Qf is efficient only if q is close to the exact flux p. Since the latter is unknown, the only way to find a suitable q is to minimize krv qk over Qf (or over a certain subspace of Qf ). Note that Z 1 1 1 krv qk2 D krvk2 f v dx C kqk2 ; 2 2 2 where the first and second terms on the right-hand side do not depend on q.

2) is not guaranteed. Also, it is worth remarking that an upper bound constructed by some q 2 Qf is efficient only if q is close to the exact flux p. Since the latter is unknown, the only way to find a suitable q is to minimize krv qk over Qf (or over a certain subspace of Qf ). Note that Z 1 1 1 krv qk2 D krvk2 f v dx C kqk2 ; 2 2 2 where the first and second terms on the right-hand side do not depend on q. Therefore, the problem is reduced to 1 inf kqk2 ; q2Qf 2 which is the dual variational problem.

Myasnikov [239]), similar estimates were derived for some classes 26 Chapter 2 Overview of convex variational problems. Estimates based on complementary principles were also obtained in A. M. Arthurth [14]. 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 [232], a strategy close to the known orthogonal projections method (see also M. Vishik [362], H. Weil [369], S.