= Céa's lemma
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{wiki}
= Céa lemma
{c}
{synonym}
For a continuous coercive bilinear form with continuity constant $M$ and coercivity constant $\alpha$, a conforming finite-element solution satisfies
$$
\|u-u_h\|_V\leq\frac M\alpha
\inf_{v_h\in V_h}\|u-v_h\|_V.
$$
For a symmetric problem measured in its <energy norm>, the constant is one.
For any $v_h\in V_h$, <Galerkin orthogonality> gives $a(e,e)=a(e,u-v_h)$, where $e=u-u_h$. Coercivity and continuity yield $\alpha\|e\|_V^2\leq M\|e\|_V\|u-v_h\|_V$. Divide by $\|e\|_V$ when it is nonzero and take the infimum. In the symmetric case, the <Cauchy-Schwarz inequality> in the <energy norm> gives the constant one.
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