Second Strang estimate with a bounded extension to the comparison space

Assume Countable Choice. Let $V$ be a real or complex Hilbert space, $a:V\times V\to \mathbb{K}$ a bounded coercive sesquilinear form (bilinear over $\mathbb{R}$), and $F$ a bounded functional with the test-linearity required by $a$. Let $V_h$ be a finite-dimensional Hilbert space with norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_h$, not necessarily a subspace of $V$. Suppose the algebraic sum $X_h=V+V_h$ is equipped with a norm extending these norms, and ${\tilde{a}}_h:X_h\times V_h\to \mathbb{K}$ is a bounded sesquilinear extension of a discrete form $a_h$ with bound $M_h$. Over $\mathbb{C}$, both forms are linear in the first argument and conjugate-linear in the second; over $\mathbb{R}$, they are bilinear. Assume $a_h$ is coercive on $V_h$ with constant ${\alpha }_h>0$, and let $F_h$ be bounded on $V_h$ with the matching test-linearity. The exact and discrete equations have unique solutions $u$ and $u_h$, and

$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_h\le (1+M_h\mathrm{/}{\alpha }_h){\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_h+{\delta }_h^{\mathrm{n}\mathrm{c}}(u)\mathrm{/}{\alpha }_h,$

where ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)={\sup }_{w_h\in V_h\setminus \{0\}}\mathrm{\mid }{F}_h(w_h)-{\tilde{a}}_h(u,w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_h$ when $V_h\mathrm{\ne }\{0\}$, and ${\delta }_h^{\mathrm{n}\mathrm{c}}(u):=0$ when $V_h=\{0\}$. The estimate requires the bounded extension and does not assert that every nonconforming, penalty, or discontinuous Galerkin form has one.

For a family of these problems, suppose $M_h\le \bar{M}<\mathrm{\infty }$ and ${\alpha }_h\ge \underset{‾}{\alpha }>0$ uniformly. Writing $E_h={\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_h$, the uniform bound is $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_h\le (1+\bar{M}\mathrm{/}\underset{‾}{\alpha })E_h+{\delta }_h^{\mathrm{n}\mathrm{c}}(u)\mathrm{/}\underset{‾}{\alpha }\mathrm{.}$ Thus uniform control of $E_h$ and ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)$ gives a uniform error budget; if both tend to zero, the errors tend to zero, and if both are $O(h^p)$ for a fixed $p>0$, the error is $O(h^p)$. No rate is inferred without those approximation, residual, and stability bounds.