First Strang estimate for conforming perturbed Galerkin problems

Assume Countable Choice. Let $V$ be a real or complex Hilbert space, $a:V\times V\to \mathbb{K}$ a bounded coercive form with boundedness constant $M$ and coercivity constant $\alpha >0$, and $F:V\to \mathbb{K}$ a bounded functional linear in the real case and conjugate-linear in the complex test argument. Let $V_h\subseteq V$ be finite-dimensional, and let $a_h:V_h\times V_h\to \mathbb{K}$ and $F_h:V_h\to \mathbb{K}$ be bounded, with $F_h$ having the same test-linearity as $F$, and $a_h$ coercive with constant ${\alpha }_h>0$ in the inherited norm. Both $a$ and $a_h$ are sesquilinear (bilinear over $\mathbb{R}$), linear in their first argument and conjugate-linear in their second over $\mathbb{C}$. There are unique exact and discrete solutions $u$ and $u_h$, and

$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_V\le {\inf }_{v_h\in V_h}((1+M\mathrm{/}{\alpha }_h)\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_V+\frac{{\delta }_{a,h}(v_h)+{\delta }_{F,h}}{{\alpha }_h}),$

where the defects are those of Variational-crime settings and consistency defects. For a family of meshes this is a uniform estimate whenever ${\inf }_h{\alpha }_h>0$ and the displayed approximation and defect terms are uniformly controlled. The term ${\delta }_{F,h}$ retains source approximation and data oscillation.