Cea's lemma
/ say-AH /
You build a finite element approximation and a natural worry surfaces: how far is my computed answer from the true solution? Cea's lemma gives a remarkably reassuring answer. It says, in plain terms, that the Galerkin finite element solution is essentially the BEST approximation the chosen mesh and polynomial space can possibly produce — up to a fixed constant, you cannot do better, and the method automatically finds something close to that best.
Here is the precise statement. Let u be the true weak solution and u_h the finite element solution from a subspace V_h. Cea's lemma states that the error in the energy norm satisfies norm(u - u_h) <= (C/alpha) times the minimum over all v_h in V_h of norm(u - v_h), where C is the boundedness constant and alpha the coercivity constant of the bilinear form. The right-hand side is the best approximation error — the distance from the true solution to the closest function the space contains. So the FEM error is at most a constant times the unavoidable best-possible error. The proof rests on Galerkin orthogonality: the error u - u_h is orthogonal (in the bilinear form) to the whole subspace V_h, which is the discrete echo of the fact that the FEM solution is the projection of the truth. Combine this quasi-optimality with standard interpolation estimates (which say how well piecewise polynomials of degree p approximate a smooth function), and you get concrete convergence rates: error proportional to h^p in the energy norm as the mesh size h shrinks.
Why it matters: Cea's lemma is what converts the abstract existence guarantee of Lax–Milgram into a usable, quantitative error bound — it is the theoretical reason FEM is trustworthy and the foundation of all a priori error analysis. Two honest caveats. First, the constant C/alpha can be large for badly-conditioned (nearly non-coercive) problems, so quasi-optimal does not mean tight. Second, it requires coercivity and the same boundedness used in Lax–Milgram; for non-coercive problems (saddle-point systems, mixed methods) you need the stronger inf-sup / Babuska–Brezzi theory instead.
Solving -u_xx = f with piecewise-linear elements (p = 1), Cea plus interpolation gives an energy-norm error proportional to h: halve the mesh and the error roughly halves. Switch to piecewise-quadratics (p = 2) and the same lemma predicts error proportional to h^2 — halving the mesh now cuts error to a quarter.
The FEM error is, up to a constant, the best the polynomial space can do.
Quasi-optimality assumes the true solution is smooth enough to be well-approximated; near a re-entrant corner or a singularity the best-approximation error itself is large, so convergence stalls — which is exactly the situation adaptive mesh refinement is built to rescue.