Lemme de Céa

Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Let be a real Hilbert space with the norm Let be a bilinear form with the properties for some constant and all in (continuity) for some constant and all in (coercivity or -ellipticity). Let be a bounded linear operator. Consider the problem of finding an element in such that for all in Consider the same problem on a finite-dimensional subspace of so, in satisfies for all in By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that for all in That is to say, the subspace solution is "the best" approximation of in up to the constant The proof is straightforward for all in We used the -orthogonality of and which follows directly from for all in . Note: Céa's lemma holds on complex Hilbert spaces also, one then uses a sesquilinear form instead of a bilinear one. The coercivity assumption then becomes for all in (notice the absolute value sign around ). In many applications, the bilinear form is symmetric, so for all in This, together with the above properties of this form, implies that is an inner product on The resulting norm is called the energy norm, since it corresponds to a physical energy in many problems. This norm is equivalent to the original norm Using the -orthogonality of and and the Cauchy–Schwarz inequality for all in . Hence, in the energy norm, the inequality in Céa's lemma becomes for all in (notice that the constant on the right-hand side is no longer present). This states that the subspace solution is the best approximation to the full-space solution in respect to the energy norm. Geometrically, this means that is the projection of the solution onto the subspace in respect to the inner product (see the adjacent picture). Using this result, one can also derive a sharper estimate in the norm . Since for all in , it follows that for all in .