Céa's lemma
/ say-AH /
The Galerkin method finds the BEST approximation a subspace can offer in the energy norm — but 'best in energy' is a strange yardstick if you care about ordinary accuracy. Céa's lemma is the reassuring bridge: it proves that the finite element solution is essentially AS GOOD as the very best the chosen shape functions could possibly do, off by at most a known constant factor. It converts the abstract Galerkin-orthogonality property into a concrete promise about accuracy.
Precisely, for a problem whose bilinear form a is bounded (with constant M) and coercive (with constant alpha), Céa's lemma states that the error of the Galerkin solution u_h satisfies ||u - u_h|| <= (M / alpha) * inf over v_h in V_h of ||u - v_h||. Read the right-hand side carefully: the infimum is the error of the BEST POSSIBLE approximation of u in the subspace V_h — the closest any shape-function combination can get. So Céa says: your computed u_h is within a fixed factor M/alpha of optimal. This is called QUASI-OPTIMALITY. The practical consequence is that finite element convergence rates reduce to a pure APPROXIMATION question: how well can piecewise polynomials of degree p on a mesh of size h approximate u? Interpolation theory answers that with O(h^p) for smooth u — and Céa's lemma immediately transfers that rate to the actual finite element error.
Céa's lemma is the cornerstone of introductory finite element error theory, but its honesty is in the constant M/alpha. For symmetric coercive problems the factor is modest and the method is genuinely near-optimal. But for problems where coercivity is weak — advection-dominated transport, where alpha is tiny — the constant M/alpha can be enormous, and the bound, though still true, becomes nearly useless: the Galerkin solution can be far from the true one (the spurious oscillations that motivate upwinding and stabilized methods). So Céa's lemma guarantees quasi-optimality, not optimality, and that gap is exactly where the hard, interesting problems live.
For -u'' = f with linear elements on a smooth solution, interpolation theory gives best-approximation error O(h) in the energy norm. Céa's lemma then guarantees the actual FEM error is also O(h), and O(h^2) in the L2 norm by a duality (Aubin-Nitsche) argument. The convergence rate of the method follows entirely from how well hats approximate u.
FEM error <= (M/alpha) x best-possible approximation error — convergence reduces to approximation theory.
Céa's lemma proves quasi-optimality, not optimality: the constant M/alpha can blow up when the problem is barely coercive (strong advection), making the bound true but loose and the plain Galerkin solution oscillatory. That regime is what stabilized and upwind methods are for.