= Solution
{c}
Take $V_h=\mathcal U_h\times\mathcal W_h\subset V$. The <Galerkin method> asks for $U_h=(u_h,w_h)\in V_h$ such that
$$
\boxed{\begin{aligned}
(\nabla u_h,\nabla v_h)_0-(w_h,v_h)_0&=0&& (v_h\in\mathcal U_h),\\
(u_h,z_h)_0+(\nabla w_h,\nabla z_h)_0&=(f,z_h)_0&& (z_h\in\mathcal W_h).
\end{aligned}}
$$
Since a <finite-dimensional subspace is closed>, $V_h$ is itself a <Hilbert space>. The <bounded bilinear form> and <coercive bilinear form> estimates in part (a) restrict to $V_h$ with the same constants. The <Lax-Milgram theorem> therefore gives exactly one $U_h$ for every $h$.
Put $E=U-U_h=(e_u,e_w)$. Subtracting the continuous and discrete <weak formulations> gives <Galerkin orthogonality>:
$$
b(E,T_h)=0\qquad(T_h\in V_h).
$$
For any $Y_h=(v_h,z_h)\in V_h$, the difference $Y_h-U_h$ also belongs to $V_h$, so
$$
\begin{aligned}
\|\nabla e_u\|_0^2+\|\nabla e_w\|_0^2
&=b(E,E)\\
&=b(E,U-Y_h)+b(E,Y_h-U_h)\\
&=b(E,U-Y_h)\\
&\leq\|E\|_V\|U-Y_h\|_V.
\end{aligned}
$$
The last step uses the constant-one <Cauchy-Schwarz inequality> estimate from part (a). Bounding the four terms separately by a sum would obscure the precise constant requested here.
The two <minimization problems> over $\mathcal U_h$ and $\mathcal W_h$ are independent. Each has a <best approximation> because its <vector subspace> is finite-dimensional. Consequently, if
$$
D_h^2=\min_{v_h\in\mathcal U_h}\|u-v_h\|_1^2+\min_{z_h\in\mathcal W_h}\|w-z_h\|_1^2,
$$
then the preceding inequality gives exactly
$$
\boxed{\|\nabla(u-u_h)\|_0^2+\|\nabla(w-w_h)\|_0^2\leq\|U-U_h\|_V D_h.}
$$
Both errors belong to the <zero-boundary Sobolev space>. Applying the <Poincaré inequality> componentwise now yields
$$
\frac1{1+K}\|U-U_h\|_V^2\leq\|U-U_h\|_V D_h.
$$
When $U=U_h$ the desired estimate is automatic; otherwise divide by the nonzero <norm>. Thus
$$
\boxed{\left(\|u-u_h\|_1^2+\|w-w_h\|_1^2\right)^{1/2}\leq(1+K)\left(\min_{v_h\in\mathcal U_h}\|u-v_h\|_1^2+\min_{z_h\in\mathcal W_h}\|w-z_h\|_1^2\right)^{1/2}.}
$$
The constant $C=1+K$ is independent of $h$. This is the <Céa lemma> estimate for this particular <skew-coupled elliptic system>.
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