Aubin–Nitsche L² error estimate from adjoint approximation

Assume Countable Choice and the conforming Galerkin setting of Céa quasi-optimality for coercive conforming Galerkin problems. Let $H=L^2(\mathrm{\Omega };\mathbb{K})$ with inner product linear in the first slot, and let $J:V\to H$ be a continuous linear inclusion with $\mathrm{\parallel }Jv{\mathrm{\parallel }}_H\le C_E\mathrm{\parallel }v{\mathrm{\parallel }}_V$. Put $e:=J(u-u_h)$. The adjoint solution $z_e$ is the unique element of $V$ satisfying $a(v,z_e)=(Jv,e{)}_H(v\in V);$ it exists by Well-posedness of the adjoint variational problem. Let $s\in (0,1]$, and let $R_s\subseteq V$ be a specified normed space of adjoint-regular functions. Assume constants $C_R,C_I$ independent of $h$ and the error satisfy $\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{R_s}\le C_R\mathrm{\parallel }e{\mathrm{\parallel }}_H,{\inf }_{w_h\in V_h}\mathrm{\parallel }{z}_e-w_h{\mathrm{\parallel }}_V\le C_I{h}^s\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{R_s}\mathrm{.}$ Then $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le M{C}_I{C}_R{h}^s\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_V\mathrm{.}$ The symbol $R_s$ means exactly the chosen regularity norm space in the two displayed assumptions; no fractional Sobolev interpolation estimate is asserted unless it verifies the second inequality. The integer endpoint may be instantiated with $R_1=H^2(\mathrm{\Omega })\cap V$, using the $H^2(\mathrm{\Omega })=W^{2,2}(\mathrm{\Omega })$ norm. If $H^2(\mathrm{\Omega })\subseteq V$, this is simply $R_1=H^2(\mathrm{\Omega })$.