Céa's lemma

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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[edit]

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

in

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 ).

Error estimate in the energy norm[edit]

The subspace solution is the projection of onto the subspace in respect to the inner product .

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 picture on the right).

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

for all in ,

it follows that

for all in .

An application of Céa's lemma[edit]

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

A string with fixed endpoints under the influence of a force pointing down.

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 string under the influence of a force such that at every point between and the force density is (where is a unit vector pointing vertically, while the endpoints of the string are on a horizontal line, see the picture on the right). For example, that force may be the gravity, when is a constant function (since the gravitational force is the same at all points).

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

for all and in

After multiplying the original boundary value problem by in this space and performing an integration by parts, one obtains the equivalent problem

for all in

with

(here the bilinear form is given by the same expression as the inner product, this is not always the case), and

It can be shown that the bilinear form and the operator satisfy the assumptions of Céa's lemma.

A function in (in red), and the typical collection of basis functions in (in blue).

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 (such functions are called piecewise-linear). 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 (the number of points in the partition that are not endpoints).

Let be the solution to the subspace problem

for all in

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

for all in

To explicitly calculate the error between and consider the function in that has the same values as at the nodes of the partition (so is obtained by linear interpolation on each interval from the values of at interval's endpoints). 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 (it depends only on the bilinear form, which implicitly depends on the interval ).

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

References[edit]

  • Johnson, Claes (1987). Numerical solution of partial differential equations by the finite element method. Cambridge University Press. ISBN 0-521-34514-6. 
  • Monk, Peter (2003). Finite element methods for Maxwell's equations. Oxford University Press. ISBN 0-19-850888-3. 
  • Roos, H.-G.; Stynes, M.; Tobiska, L. (1996). Numerical methods for singularly perturbed differential equations: convection-diffusion and flow problems. Berlin; New York: Springer-Verlag. ISBN 3-540-60718-8. 
  • Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C. (1996). Computational differential equations. Cambridge; New York: Cambridge University Press. ISBN 0-521-56738-6. 
  • Zeidler, Eberhard (1995). Applied functional analysis: applications to mathematical physics. New York: Springer-Verlag. ISBN 0-387-94442-7.