Céa's lemma

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.

be a real Hilbert space with the norm

be a bounded linear operator.

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

Note: Céa's lemma holds on complex Hilbert spaces also, one then uses a sesquilinear form

(notice the absolute value sign around

is symmetric, so This, together with the above properties of this form, implies that

The resulting norm is called the energy norm, since it corresponds to a physical energy in many problems.

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).

Using this result, one can also derive a sharper estimate in the norm

Since it follows that We will apply Céa's lemma to estimate the error of calculating the solution to an elliptic differential equation by the finite element method.

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

is a unit vector pointing vertically, while the endpoints of the string are on a horizontal line, see the adjacent picture).

is a constant function (since the gravitational force is the same at all points).

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

satisfy the assumptions of Céa's lemma.

In order to determine a finite-dimensional subspace

be the space of all continuous functions that are affine on each subinterval in the partition (such functions are called piecewise-linear).

(the number of points in the partition that are not endpoints).

as of a piecewise-linear approximation to the exact solution

By Céa's lemma, there exists a constant

is obtained by linear interpolation on each interval

It can be shown using Taylor's theorem that there exists a constant

This inequality then yields an estimate for the error Then, by substituting

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 (here the domain of

was in one dimension), and while using higher order polynomials for the subspace