---
tag: 4X6WC3E7P8
kind: statement
role: lemma
name: Céa's lemma
msc: [65N30, 65N15]
uses: [HM2HHE3J51, 0SQZMXB6GP, 86KRHS6X22, 4A82S5FVRT, Y5B0PEBKE8, 85KK79FCN5]
about: [Y5B0PEBKE8]
sources:
  - url: https://en.wikipedia.org/wiki/C%C3%A9a%27s_lemma
same_as:
  wikidata: Q1816912
  wikipedia: Céa's lemma
---

## Statement

Let $V$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
Let $a \colon V \times V \to \R$ be a [[bilinear form|0SQZMXB6GP]] that is bounded, $\abs{a(u, v)} \le M \norm{u} \norm{v}$ for all $u, v \in V$, and [[coercive|86KRHS6X22]], $a(v, v) \ge \alpha \norm{v}^2$ for all $v \in V$, with constants $M \ge 0$ and $\alpha > 0$.
Let $f$ be an element of the [[continuous dual space|4A82S5FVRT]] of $V$, and let $u \in V$ satisfy $a(u, v) = f(v)$ for all $v \in V$.
Let $V_h$ be a closed subspace of $V$, and let $u_h \in V_h$ satisfy $a(u_h, v) = f(v)$ for all $v \in V_h$.
Then
$$\norm{u - u_h} \le \frac{M}{\alpha} \, \inf_{v \in V_h} \norm{u - v}.$$

## Remarks

Both $u$ and $u_h$ exist and are unique by the [[Lax–Milgram theorem|85KK79FCN5]], 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.

## Explanations

### L1 human

The finite element method looks for an approximate solution inside a smaller space of simple functions.
Céa's lemma says that this approximate solution is almost as good as the best possible approximation from that space.
The loss is at most a fixed factor $M / \alpha$, which depends on the problem but not on the space.

### L2 human

The lemma splits error analysis of [[Galerkin methods|Y5B0PEBKE8]] 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|K4P0R83G1Q]].
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|SC5FV3GHV5]].