CÃ©a's lemma

The lemma of CEA or CEA- lemma is a mathematical theorem from functional analysis. It is fundamental to the error estimation of finite element approximations of elliptic partial differential equations. The lemma is named in honor of the French mathematician Jean Cea, who proved it in his dissertation in 1964.

2.1 The energy norm
2.2 The lemma of CEA in the energy norm
2.3 Conclusions

3.1 Galerkin orthogonality
3.2 assessment

Formulation

Requirements

Be a real Hilbert space with the norm. Be a bilinear form, the

Limited ( equivalently continuous), ie for a constant, and all

And coercive (often strongly positive, V -elliptic ), ie for a constant, and all

Is. Further, let a bounded linear operator.

Problem

Consider the problem with a

To find. Now consider the same problem in a subspace, ie it is to find one with

By the lemma of Lax- Milgram, there is a unique solution for both problems.

Statement of the lemma

If the above conditions are met, then the lemma states of CEA:

This means that the approximation of the solution of the sub-space is inferior to the constant only as the best approximation to the room, it is a quasi- optimal.

Comments

With a symmetric bilinear form, the constant reduced to, the proof is given below.

The lemma of CEA also applies to complex Hilbert spaces by a sesquilinear form instead of the bilinear form is used. The coercivity is then for all, note the amount of characters.

The approximation ratio of the approximation error approach area determined strong.

Special case: Symmetric bilinear form

The energy norm

In many applications, the bilinear form is symmetric, therefore for all. With the requirements of the CEA- lemma implies that a scalar product of being. The implied standard is called the energy norm, because in many physical problems is an energy. This standard is equivalent to the norm of the vector space.

The lemma of CEA in the energy norm

From the Galerkin orthogonality of and with the Cauchy- Schwarz inequality arises

Thus, the lemma of CEA is in the energy norm:

Note that the constant is on the right side disappears.

This means that the sub-space solution is the best approximation of the solution with respect to the energy standard. Geometric can be interpreted as a projection of respect to the subspace.

Conclusions

This allows the sharper bound for symmetric bilinear forms show for the usual norm of the vector space. from

Follows

Evidence

The proof is long and leads the necessity of the conditions in mind.

Galerkin orthogonality

The guidance given in the problem equation for all and for all be subtracted from each other, which is possible because. The resulting equation is for all and is called Galerkin orthogonality.