Let be a bounded linear operator. Consider the problem of finding an element in such that Consider the same problem on a finite-dimensional subspace of so, in satisfies By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that That is to say, the subspace solution is "the best" approximation of in up to the constant The proof is straightforward We used the -orthogonality of and which follows directly from Note: Céa's lemma holds on complexHilbert spaces also, one then uses a sesquilinear form instead of a bilinear one. The coercivity assumption then becomes for all in .
In many applications, the bilinear form is symmetric, so 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 Hence, in the energy norm, the inequality in Céa's lemma becomes . 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 . Using this result, one can also derive a sharper estimate in the norm. Since it follows that
An application of Céa's lemma
We will apply Céa's lemma to estimate the error of calculating the solution to an elliptic differential equation by the finite element method. Consider the problem of finding a function satisfying the conditions where is a given continuous function. Physically, the solution to this two-point boundary value problem represents the shape taken by a stringunder the influence of a force such that at every point between and the force density is . For example, that force may be the gravity, when is a constant function. Let the Hilbert space be the Sobolev space which is the space of all square-integrable functions defined on that have a weak derivative on with also being square integrable, and satisfies the conditions The inner product on this space is After multiplying the original boundary value problem by in this space and performing an integration by parts, one obtains the equivalent problem with , and It can be shown that the bilinear form and the operator satisfy the assumptions of Céa's lemma. In order to determine a finite-dimensional subspace of consider a partition of the interval and let be the space of all continuous functions that are affine on each subinterval in the partition. In addition, assume that any function in takes the value 0 at the endpoints of It follows that is a vector subspace of whose dimension is . Let be the solution to the subspace problem so one can think of as of a piecewise-linear approximation to the exact solution By Céa's lemma, there exists a constant dependent only on the bilinear form such that To explicitly calculate the error between and consider the function in that has the same values as at the nodes of the partition. It can be shown using Taylor's theorem that there exists a constant that depends only on the endpoints and such that for all in where is the largest length of the subintervals in the partition, and the norm on the right-hand side is the L2 norm. This inequality then yields an estimate for the error Then, by substituting in Céa's lemma it follows that where is a different constant from the above. This result is of a fundamental importance, as it states that the finite element method can be used to approximately calculate the solution of our problem, and that the error in the computed solution decreases proportionately to the partition size Céa's lemma can be applied along the same lines to derive error estimates for finite element problems in higher dimensions, and while using higher order polynomials for the subspace
