= Solution
{c}
Use the continuous <piecewise-linear> nodal <interpolants> $I_hu$ and $I_hw$ on the <shape-regular mesh>. Since $u,w\in H^2(\Omega)$ in two dimensions, the <Sobolev embedding theorem> supplies continuous representatives, so the nodal values are well-defined. Their zero <Sobolev traces> imply zero boundary nodal values. Hence $I_hu\in\mathcal U_h$ and $I_hw\in\mathcal W_h$ for the usual full <conforming finite element spaces>.
The permitted <finite element interpolation estimates> give constants $C_u,C_w$ determined by the <shape regularity> and $\Omega$, independent of $h$, such that
$$
\|u-I_hu\|_1\leq C_u h\|u\|_{H^2(\Omega)},\qquad
\|w-I_hw\|_1\leq C_w h\|w\|_{H^2(\Omega)}.
$$
Substitute these choices in the <best approximation> bound from part (b):
$$
\boxed{\begin{aligned}
\left(\|u-u_h\|_1^2+\|w-w_h\|_1^2\right)^{1/2}&\leq C_*h,\\
C_*&=(1+K)\left(C_u^2\|u\|_{H^2(\Omega)}^2+C_w^2\|w\|_{H^2(\Omega)}^2\right)^{1/2}.
\end{aligned}}
$$
For fixed $u,w$, this constant is independent of $h$. A <shape-regular mesh> suffices; no uniformity of the individual element sizes is needed, since $h$ is their maximum diameter.
The inclusion of these <interpolants> is an essential assumption. If the wording “consist of” were read as permitting arbitrary smaller <vector subspaces>, taking $\mathcal U_h=\mathcal W_h=\{0\}$ for every $h$ would prevent the claimed convergence for a nonzero solution. The full <conforming finite element space> interpretation is also needed for the dimension assertion in part (d).
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