Céa's lemma

Remarks

Both $u$ and $u_h$ exist and are unique by the Lax–Milgram theorem, applied on $V$ and on $V_h$. Subtracting the two equations gives Galerkin orthogonality: $a(u - u_h, w) = 0$ for all $w \in V_h$. For $v \in V_h$, take $w = v - u_h$: then $\alpha \norm{u - u_h}^2 \le a(u - u_h, u - u_h) = a(u - u_h, u - v) \le M \norm{u - u_h} \norm{u - v}$. Dividing by $\alpha \norm{u - u_h}$ and taking the infimum over $v$ gives the claim. If $a$ is also symmetric, then $u_h$ is the $a$-orthogonal projection of $u$ onto $V_h$, and the constant improves to $\sqrt{M / \alpha}$.

Example: for $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $V = H^1_0(0, 1)$, $a(u, v) = \int_0^1 u' v' \, dx$, and $V_h$ the continuous piecewise linear functions on a mesh of width $h$. Choosing $v$ to be the piecewise linear interpolant of $u$ in the infimum gives $\norm{u - u_h}_{H^1} \le C h \norm{u''}_{L^2}$ when $u \in H^2(0, 1)$.

Coercivity is needed. Take an indefinite form, such as $a(u, v) = \int_0^1 (u' v' - k^2 u v) \, dx$ with $k$ large. Then a discrete solution may fail to exist on a given $V_h$. When it exists, it need not be close to the best approximation unless $h$ is small enough.

Going deeper

For a student

The lemma splits error analysis of Galerkin methods into two independent questions. One is stability, measured by $M / \alpha$, which comes from the continuous problem. The other is approximation: how well functions in $V_h$ approximate the exact solution, which is a question of interpolation theory and is answered with the Bramble–Hilbert lemma. For problems that are not coercive, such as saddle-point problems, the role of $\alpha$ is taken by an inf-sup constant, as in the Ladyzhenskaya–Babuška–Brezzi condition.