Then n 1 the minimum of i w and au w u w 8w 2 s0

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Unformatted text preview: d by the same function U . N 2. The function U is the orthogonal projection of u onto S0 with respect to strain energy, i.e., A(V u ; U ) = 0 N 8V 2 S0 : (7.2.6) N 3. The minimizing function U 2 S0 satis es the Galerkin problem A(V U ) = (V f ) N 8V 2 S0 : (7.2.7) 1 8v 2 H0 : (7.2.8) N 1 In particular, if S0 is the whole of H0 A(v u) = (v f ) Proof. Our proof will omit several technical details, which appear in, e.g., Wait and Mitchell 21], Chapter 6. N Let us begin with (7.2.7). If U minimizes I W ] over S0 then for any and any N V 2 S0 I U ] I U + V ]: Using (7.2.5), I U ] A(U + V U + V ) ; 2(U + V f ) or I U ] I U ] + 2 A(V U ) ; (V f )] + 2 A(V V ) or 0 2 A(V U ) ; (V f )] + 2 A(V V ): This inequality must hold for all possible of either sign thus, (7.2.7) must be satis ed. N Equation (7.2.8) follows by repeating these arguments with S0 replaced by H01. N Next, replace v in (7.2.8) by V 2 S0 H01 and subtract (7.2.7) to obtain (7.2.6). In order to prove Conclusion 1, consider the identity A(u ; U...
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