Céa quasi-optimality for coercive conforming Galerkin problems

Apply [F4] on $V$; its hypotheses are exactly the boundedness, coercivity, Hilbert, and datum assumptions in Given. This gives a unique $u\in V$ with $a(u,v)=F(v)$ for all $v\in V$.

Fix a finite ordered basis of $V_h$. By [F3], $V_h$ is closed in $V$. A Cauchy sequence in $V_h$ is Cauchy in $V$, so [F2] gives a limit in $V$; the closedness puts the limit in $V_h$. The restricted inner product is complete, so $V_h$ is Hilbert, also when $V_h=\{0\}$.

The restrictions of $a$ and $F$ to $V_h$ retain the same bounds $M$ and $\alpha$, and $F{\mathrm{\mid }}_{V_h}$ is bounded. Apply [F4] on the Hilbert space from step 1.2. This gives a unique $u_h\in V_h$ satisfying the discrete equations.

By [F5], with $e:=u-u_h$, we have $a(e,w_h)=0$ for every $w_h\in V_h$.

Fix any $v_h\in V_h$. Since $u-v_h=e+(u_h-v_h)$ and $u_h-v_h\in V_h$, [F5] gives $a(e,e)=a(e,u-v_h)-a(e,u_h-v_h)=a(e,u-v_h)\mathrm{.}$ Coercivity and boundedness therefore give $\alpha \mathrm{\parallel }e{\mathrm{\parallel }}_V^2\le \mathrm{Re}a(e,e)\le \mathrm{\mid }a(e,u-v_h)\mathrm{\mid }\le M\mathrm{\parallel }e{\mathrm{\parallel }}_V\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_V\mathrm{.}$

If $e=0$, the claimed estimate is immediate. Otherwise divide step 4.1 by $\alpha \mathrm{\parallel }e{\mathrm{\parallel }}_V$ and then take the infimum over all $v_h\in V_h$. This proves Céa's estimate.