Past exam of the finite element methods for partial differential equations course of the University of Oxford / 2023 / 1 / c / Solution

Use the continuous piecewise-linear nodal interpolants $I_{h}u$ and $I_{h}w$ 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 $I_{h}u\in {\mathcal{U}}_h$ and $I_{h}w\in {\mathcal{W}}_h$ 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 thatSubstitute these choices in the best approximation bound from part (b):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).