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.

Lemma statement

Let V be a real Hilbert space with the norm |\cdot|. Let be a bilinear form with the properties Let be a bounded linear operator. Consider the problem of finding an element u in V such that Consider the same problem on a finite-dimensional subspace V_h of V, so, u_h in V_h 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 u_h is "the best" approximation of u in V_h, up to the constant The proof is straightforward We used the a-orthogonality of u-u_h and which follows directly from 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 v in V (notice the absolute value sign around a(v, v)).

Error estimate in the energy norm

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 V. 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 |\cdot|. Using the a-orthogonality of u-u_h and V_h and the Cauchy–Schwarz inequality Hence, in the energy norm, the inequality in Céa's lemma becomes (notice that the constant on the right-hand side is no longer present). This states that the subspace solution u_h is the best approximation to the full-space solution u in respect to the energy norm. Geometrically, this means that u_h is the projection of the solution u onto the subspace V_h in respect to the inner product (see the adjacent picture). Using this result, one can also derive a sharper estimate in the norm | \cdot |. 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 u to this two-point boundary value problem represents the shape taken by a string under the influence of a force such that at every point x between a and b the force density is (where \mathbf{e} is a unit vector pointing vertically, while the endpoints of the string are on a horizontal line, see the adjacent picture). For example, that force may be the gravity, when f is a constant function (since the gravitational force is the same at all points). Let the Hilbert space V be the Sobolev space which is the space of all square-integrable functions v defined on [a, b] that have a weak derivative on [a, b] with v' also being square integrable, and v satisfies the conditions The inner product on this space is After multiplying the original boundary value problem by v 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 L satisfy the assumptions of Céa's lemma. In order to determine a finite-dimensional subspace V_h of V, consider a partition of the interval [a, b], and let V_h be the space of all continuous functions that are affine on each subinterval in the partition (such functions are called piecewise-linear). In addition, assume that any function in V_h takes the value 0 at the endpoints of [a, b]. It follows that V_h is a vector subspace of V whose dimension is n-1 (the number of points in the partition that are not endpoints). Let u_h be the solution to the subspace problem so one can think of u_h as of a piecewise-linear approximation to the exact solution u. By Céa's lemma, there exists a constant C>0 dependent only on the bilinear form such that To explicitly calculate the error between u and u_h, consider the function \pi u in V_h that has the same values as u at the nodes of the partition (so \pi u is obtained by linear interpolation on each interval from the values of u at interval's endpoints). It can be shown using Taylor's theorem that there exists a constant K that depends only on the endpoints a and b, such that for all x in [a, b], where h 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 v=\pi u in Céa's lemma it follows that where C is a different constant from the above (it depends only on the bilinear form, which implicitly depends on the interval [a, b]). 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 h. Céa's lemma can be applied along the same lines to derive error estimates for finite element problems in higher dimensions (here the domain of u was in one dimension), and while using higher order polynomials for the subspace V_h.