Unformatted text preview: ; V u ; U ; V ) = A(u ; U u ; U ) ; 2A(u ; U V ) + A(V V ): 7.2. Convergence and Optimality 7 Using (7.2.6) A(u ; U u ; U ) = A(u ; U ; V u ; U ; V ) ; A(V V ):
Since A(V V ) 0, A(u ; U u ; U ) A(u ; U ; V u ; U ; V ) N
8V 2 S0 : Equality only occurs when V = 0 therefore, U is the unique minimizing function.
Remark 2. We proved a similar result for one-dimensional problems in Theorems
Remark 3. Continuity and coercivity did not appear in the proof however, they are
needed to establish existence, uniqueness, and completeness. Thus, we never proved that
limN !1 U = u. A complete analysis appears in Wait and Mitchell 21], Chapter 6.
Remark 4. The strain energy A(v u) not need be symmetric. A proof without this
restriction appears in Ciarlet 13]. Corollary 7.2.1. With the assumptions of Theorem 7.2.2,
A(u ; U u ; U ) = A(u u) ; A(U U ): (7.2.9) Proof. cf. Problem 3 at the end of this section. In Section 4.6, we obtained a priori estimates of interpolation errors under some
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