By Weimin Han

This quantity presents a posteriori mistakes research for mathematical idealizations in modeling boundary worth difficulties, particularly these coming up in mechanical functions, and for numerical approximations of various nonlinear variational difficulties. the writer avoids giving the consequences within the such a lot basic, summary shape in order that it truly is more uncomplicated for the reader to appreciate extra essentially the basic principles concerned. Many examples are incorporated to teach the usefulness of the derived mistakes estimates.

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**Example text**

Sobolev embedding theorems. Sobolev embedding theorems are important, especially in analyzing the regularity of a weak solution of a boundary value problem. 1 1 Let V and W be two Banach spaces with V C W . We say the space V is continuously embedded in W and write V v W ifthere exists a constant c > 0 such that 14 A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY We say the space V is compactly embedded in W and write V 3- W , if ( 1 . 5 ) holds and each bounded sequence in V has a convergent subsequence in W.

6). 26), characterized by the relation Let us try to determine this best constant, Preliminaries First we have an existence result for the best constant. 17 There exists 0 f u E H : ~( R )such that Proof. , m. 13 for the compact embedding of H& ( 0 ) in L ' ( R ) to conclude the existence of a subsequence {un,} and u E H+l ( 0 ) such that Thus we have (cf. 1)) Therefore, . By the definition of the constant co, the above inequality is an equality. Now we perform a formal calculation to derive a formula for the computation of co and then prove the formula rigorously.

2 1 Let d > 2. 27) is the weak form of the boundary value problem: < a~gi. together with the interface condition that both u ( x ) and d u ( x ) / d l x l are continuous across the sphere 1x1 = ro. For d = 2, the solution is For d > 2, the solution is "I2 -- u(x) = + 2d rf d (d - 2) 3 2d + o'd d ( d - 2) (ri-d-~i-d) i f 0 5 1x1 < T O , ~i-~) ( 1 ~ l 2 - ~ i f r o 5 1x15 RO A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY 24 and so Here r ( . ) denotes the gamma function (cf. , [104]). Hence, we have the optimal inequality and In particular, when ro = Ro, we have the optimal inequality for any dimension d 2 2.