The h^(r+s) L² rate under primal and dual approximation

Assume Countable Choice. Assume the real scalar finite-element setting of Primal H¹ approximation on shape-regular affine finite-element meshes, and set $V:=H_0^1(\mathrm{\Omega })$ with its usual $H^1$ norm. Let $V_h\subseteq V$ be the stated conforming discrete space, let $a:V\times V\to \mathbb{R}$ be the same bounded coercive form as in Céa quasi-optimality for coercive conforming Galerkin problems, and let $u\in W^{r+1,2}(\mathrm{\Omega })\cap V$ and $u_h\in V_h$ solve the continuous and discrete equations for the same bounded linear datum $F:V\to \mathbb{R}$; assume also that $V_h$ is finite-dimensional. The natural inclusion $J:V\to L^2(\mathrm{\Omega })$ obeys $\mathrm{\parallel }Jv{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$. For $e:=J(u-u_h)$, let $z_e$ be the adjoint solution from Aubin–Nitsche L² error estimate from adjoint approximation. Suppose there are $s\in (0,1]$, a normed regularity space $R_s\subseteq V$ for $z_e$, and constants $C_D,C_R$ independent of $h$ and $e$ such that $\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{R_s}\le C_R\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\mathrm{.}$ ${\inf }_{w_h\in V_h}\mathrm{\parallel }{z}_e-w_h{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}\le C_D{h}^s\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{R_s}\mathrm{.}$ Then $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le \frac{M^2}{\alpha }{C}_{\mathrm{F}\mathrm{E}}{C}_D{C}_R{h}^{r+s}\mathrm{\mid }u{\mathrm{\mid }}_{W^{r+1,2}(\mathrm{\Omega })}\mathrm{.}$ Here $C_{\mathrm{F}\mathrm{E}}$ is the primal interpolation constant and $M,\alpha$ are the boundedness and coercivity constants of $a$. In a full-adjoint Poisson branch, one may take $R_1=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ with the $H^2$ norm, provided both displayed dual bounds hold. The $h^{r+1}$ rate is conditional on those bounds and the primal approximation bound; no full-adjoint regularity is asserted on an arbitrary Lipschitz domain.

The $h^{r+1}$ conclusion requires a dual approximation estimate of order one for the actual discrete spaces; the primal interpolation item alone does not establish that estimate for every degree and every degree-of-freedom family. A separate global regularity input must also match the actual domain and adjoint operator. The rate is not unconditional on every polyhedral or Lipschitz problem.