Céa's_lemma

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Céa's lemmaFrom Wikipedia, the free encyclopedia

Céa's lemma is a lemma in mathematics. It is an important tool for proving error estimates for the finite element method applied toelliptic partial differential equations.

Contents

1 Lemma statement2 Error estimate in the energy norm3 An application of Céa's lemma4 References

Lemma statement

Let V be a real Hilbert space with the norm Let be a bilinear form with the properties

for some constant γ > 0 and all v,w in V (continuity)

for some constant α > 0 and all v in V (coercivity or V ellipticity).

Let be a bounded linear operator. Consider the problem of finding an element u in V such that

for all v in

Consider the same problem on a finite dimensional subspace V h of V , so, uh in V h satisfies

for all v in

By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that

for all v in V h.

That is to say, the subspace solution uh is "the best" approximation of u in V h, up to the constant γ / α.

The proof is straightforward

for all v in V h.

We used the a orthogonality of u − uh and V h

in V h

which follows directly from

a(u,v) = L(v) = a (uh,v) for all v in V h.

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

lemma Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Céa's_lemm

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The subspace solution uh is the projection of u onto the subspace V hin respect to the inner product .

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

, .

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. Thisnorm is equivalent to the original norm

Using the a orthogonality of u − uh and V h and the Cauchy–Schwarz inequality

for all v in V h

Hence, in the energy norm, the inequality in Céa's lemma becomes

for all v in V h

(notice that the constant γ / α on the right hand side is no longer present).

This states that the subspace solution uh is the best approximation to the full space solution u in respect to the energy norm.Geometrically, this means that uh is the projection of the solution u onto the subspace V h in respect to the inner product (seethe picture on the right).

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

for all v in V h,

it follows that

for all v in V h.

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 elementmethod.

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 is a unitvector pointing vertically, while the endpoints of the string are on a horizontal line, see the picture on the right). For example, that forcemay 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 v(a) = v(b) = 0. The inner producton this space is

for all v and w in


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A function in V h (in red), and thetypical collection of basis functions inV h (in blue).

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

for all v in V ,

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 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 thatany 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 notendpoints).

Let uh be the solution to the subspace problem

for all v in V h,

so one can think of uh as of a piecewise linear approximation to the exact solution u. By Céa's lemma, there exists a constant C > 0dependent only on the bilinear form such that

for all v in

To explicitly calculate the error between u and uh, consider the function πu in V h that has the same values as u at the nodes of the partition (so πu is obtained by linear interpolation on each interval [ xi, xi + 1] from the values of u at interval's endpoints). It can beshown 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 [ xi, xi + 1] in the partition, and the norm on the right hand side is theL2 norm.

This inequality then yields an estimate for the error

Then, by substituting v = πu 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 solutionof our problem, and that the error in the computed solution decreases proportionately to the partition size h. Céa's lemma can be appliedalong the same lines to derive error estimates for finite element problems in higher dimensions (here the domain of u was in onedimension), and while using higher order polynomials for the subspace V h.


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