Lemma of Céa

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

formulation

requirements

Let be a real Hilbert space with the norm . Be a bilinear form that ${\displaystyle V}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle a\colon V\times V\to \mathbb {R} }$

• bounded (equivalent to continuous), d. H. for a constant and all ${\displaystyle |a(v,w)|\leq C\|v\|\,\|w\|}$ ${\displaystyle C>0}$ ${\displaystyle v,w\in V}$

• and coercive (often also strongly positive, V-elliptical), d. H. for a constant and all ${\displaystyle a(v,v)\geq \alpha \|v\|^{2}}$ ${\displaystyle \alpha >0}$ ${\displaystyle v\in V}$

is. Let further be a bounded linear operator . ${\displaystyle L\colon V\to \mathbb {R} }$

Problem

Consider the problem, one with ${\displaystyle u\in V}$

${\displaystyle a(u,v)=L(v)\,}$ for all ${\displaystyle v\in V}$

to find. Now consider the same problem in a subspace , i. H. there is one to find with ${\displaystyle V_{h}\subset V}$ ${\displaystyle u_{h}\in V_{h}}$

${\displaystyle a(u_{h},v)=L(v)\,}$for everyone . ${\displaystyle v\in V_{h}}$

According to Lax-Milgram's lemma, there is a clear solution to both problems.

Statement of the lemma

If the above conditions are met, then Céa's lemma says :

${\displaystyle \|u-u_{h}\|\leq {\frac {C}{\alpha }}\inf _{v_{h}\in V_{h}}\left(\|u-v_{h}\|\right)}$.

This means that the approximation of the solution from the subspace is at most by the constant worse than the best approximation for in the space , it is quasi-optimal . ${\displaystyle u_{h}}$ ${\displaystyle V_{h}}$ ${\displaystyle {\tfrac {C}{\alpha }}}$ ${\displaystyle u}$ ${\displaystyle V_{h}}$

Remarks

With a symmetrical bilinear form the constant decreases to , the proof is given below. ${\displaystyle \textstyle {\sqrt {C/\alpha }}}$

Céa's lemma also applies to complex Hilbert spaces by using a sesquilinear form instead of the bilinear form. The coercivity then becomes for everyone , note the symbols around . ${\displaystyle a(\cdot ,\cdot )}$ ${\displaystyle |a(v,v)|\geq \alpha \|v\|^{2}}$ ${\displaystyle v\in V}$ ${\displaystyle a(v,v)}$

The approximation quality of the approach space strongly determines the approximation error . ${\displaystyle V_{h}}$ ${\displaystyle \|u-u_{h}\|}$

Special case: symmetrical bilinear form

The energy standard

In many applications the bilinear form is symmetrical, i.e. for all in . With the assumptions of the Céa lemma it follows that is a scalar product of . The implied norm is called the energy norm because it represents an energy in many physical problems . This norm is equivalent to the norm of the vector space . ${\displaystyle a}$ ${\displaystyle a(v,w)=a(w,v)}$ ${\displaystyle v,w}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle V}$ ${\displaystyle \|v\|_{a}={\sqrt {a(v,v)}}}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle V}$

Céa's lemma in the energy norm

The subspace solution is a projection of onto the subspace with respect to the scalar product . ${\displaystyle u_{h}}$ ${\displaystyle u}$ ${\displaystyle V_{h}}$ ${\displaystyle a}$

The Galerkin orthogonality of mit and the Cauchy-Schwarz inequality results ${\displaystyle u-u_{h}}$ ${\displaystyle V_{h}}$

${\displaystyle \|u-u_{h}\|_{a}^{2}=a(u-u_{h},u-u_{h})=a(u-u_{h},u-v)\leq \|u-u_{h}\|_{a}\cdot \|u-v\|_{a}}$for everyone in . ${\displaystyle v}$ ${\displaystyle V_{h}}$

Thus Céa's lemma in the energy norm reads :

${\displaystyle \|u-u_{h}\|_{a}\leq \|u-v\|_{a}}$ for everyone in . ${\displaystyle v}$ ${\displaystyle V_{h}}$

Note that the constant on the right has disappeared. ${\displaystyle {\tfrac {C}{\alpha }}}$

This means that the subspace solution is the best approximation of the solution with respect to the energy norm. Geometrically can be a projection with respect of to the subspace interpret. ${\displaystyle u_{h}}$ ${\displaystyle u}$ ${\displaystyle u_{h}}$ ${\displaystyle a}$ ${\displaystyle u}$ ${\displaystyle V_{h}}$

Inferences

This shows the sharper bound for symmetric bilinear forms for the normal norm of vector space . Out ${\displaystyle \|\cdot \|}$ ${\displaystyle V}$

${\displaystyle \alpha \|u-u_{h}\|^{2}\leq a(u-u_{h},u-u_{h})=\|u-u_{h}\|_{a}^{2}\leq \|u-v\|_{a}^{2}\leq C\|u-v\|^{2}}$for everyone in ${\displaystyle v}$ ${\displaystyle V_{h}}$

follows

${\displaystyle \|u-u_{h}\|\leq {\sqrt {\frac {C}{\alpha }}}\|u-v\|}$ for everyone in . ${\displaystyle v}$ ${\displaystyle V_{h}}$

proof

The proof is not long and shows the necessity of the premises.

Galerkin orthogonality

The equations for all and for all given in the problem are subtracted from each other, which is possible because of . The resulting equation reads for all and is called Galerkin orthogonality. ${\displaystyle a(u,v)=L(v)}$ ${\displaystyle v\in V}$ ${\displaystyle a(u_{h},v)=L(v)}$ ${\displaystyle v\in V_{h}}$ ${\displaystyle V_{h}\subset V}$ ${\displaystyle a(u-u_{h},v)=0}$ ${\displaystyle v\in V_{h}}$

appraisal

The bilinear form is coercive ${\displaystyle a}$

${\displaystyle \alpha \|u-u_{h}\|^{2}\leq a(u-u_{h},u-u_{h})}$

Addition of 0, let ${\displaystyle v_{h}\in V_{h}}$

${\displaystyle =a(u-u_{h},u-v_{h}+v_{h}-u_{h})}$

With bilinearity of ${\displaystyle a}$

${\displaystyle =a(u-u_{h},u-v_{h})+a(u-u_{h},v_{h}-u_{h})}$

The second term is 0 because of the Galerkin orthogonality, da ${\displaystyle v:=v_{h}-u_{h}\in V_{h}}$

${\displaystyle =a(u-u_{h},u-v_{h})}$

The bilinear form is continuous ${\displaystyle a}$

${\displaystyle \leq C\|u-u_{h}\|\|u-v_{h}\|}$

The equation can be divided by. Since arbitrary is chosen, the infimum can also be chosen, which gives us the statement. ${\displaystyle \|u-u_{h}\|}$ ${\displaystyle v_{h}}$ ${\displaystyle V_{h}}$

literature

• D. Braess: Finite element theory, fast solvers and applications in elasticity theory . 4th edition. Springer, 2007, ISBN 978-3-540-72449-0 (Chapter II §4.2 and Chapter III §1.1).

• Jean Céa, Approximation variationnelle des problemèmes aux limites, Annales de l'institut Fourier, Volume 14, No. 2, 1964, pp. 345–444, PDF, 5 MB (original work by J. Céa)

Individual evidence

1. ↑ ^{a b} E. Emmrich: Ordinary and Operator Differential Equations - An integrated introduction to boundary value problems and evolution equations for students . 1st edition. Vieweg + Teubner, 2004, ISBN 978-3-528-03213-5 , page 112