= Aubin–Nitsche duality argument
{c}
For a symmetric elliptic variational problem with <Galerkin orthogonality>, let $e=u-u_h$ and solve the dual problem $a(v,z)=(e,v)_{L^2}$ for every $v\in H_0^1(\Omega)$. If <elliptic regularity> gives $\|z\|_{H^2}\leq C\|e\|_{L^2}$, then a <finite element interpolation estimate> gives
$$
\|e\|_{L^2}^2=a(e,z-I_hz)
\leq Ch\|e\|_{H^1}\|z\|_{H^2}
\leq Ch\|e\|_{H^1}\|e\|_{L^2}.
$$
Thus $\|e\|_{L^2}\leq Ch\|e\|_{H^1}$. Combining this with the <Céa lemma> and an $O(h)$ <energy norm> error gives an $O(h^2)$ <L2 norm> error. The additional order depends on the dual regularity assumption, which can fail on unsuitable domains.
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