= Ritz-Galerkin equivalence for a symmetric coercive form
{c}
{title2=$J(v)-J(u)=\tfrac12a(v-u,v-u)$}
For a bounded symmetric <coercive bilinear form> on a Hilbert space, a solution of $a(u,w)=\ell(w)$ minimizes $J(v)=a(v,v)/2-\ell(v)$. The displayed identity proves strict minimality and uniqueness. Restricting to a <conforming finite element space> makes the discrete <Ritz method> equivalent to the <Galerkin method>. <Galerkin orthogonality> gives exact best approximation in the <energy norm>; the <Céa lemma> translates it into a reference-norm estimate.
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