Céa's lemma
Encyclopedia
Céa's lemma is a lemma
in mathematics
. It is an important tool for proving error estimates for the finite element method
applied to elliptic
partial differential equation
s.
be a real
Hilbert space
with the norm
Let 
be a bilinear form with the properties
Let
be a bounded linear operator. Consider the problem of finding an element 
in 
such that
Consider the same problem on a finite-dimensional subspace
of 
so, 
in 
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
is "the best" approximation of 
in 
up to
the constant
The proof is straightforward
Lemma (mathematics)
In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It is an important tool for proving error estimates for the finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
applied to elliptic
Elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...
partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s.
Lemma statement
Let be a realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
with the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
Let be a bilinear form with the properties-
for some constant 
and all 
in 
(continuityContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
)
-
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 toUp to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
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 complexComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
Hilbert spaces also, one then uses a sesquilinear formSesquilinear formIn mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...
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
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 inequalityCauchy–Schwarz inequalityIn mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...
-
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 projectionProjection (linear algebra)In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
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
We will apply Céa's lemma to estimate the error of calculating the solution to an elliptic differential equation by the finite element methodFinite element methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
.
Consider the problem of finding a function
satisfying the conditions
where
is a given continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
.
Physically, the solution
to this two-point boundary value problemBoundary value problemIn mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
represents the shape taken by a stringRopeA rope is a length of fibres, twisted or braided together to improve strength for pulling and connecting. It has tensile strength but is too flexible to provide compressive strength...
under the influence of a force such that at every point
between 
and 
the force densityForce densityIn fluid mechanics, the force density is the negative gradient of pressure. It has the physical dimensions of force per unit volume. Force density is a vector field representing the flux density of the hydrostatic force within the bulk of a fluid...
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 spaceSobolev spaceIn mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
which is the space of all square integrable functions 
defined on 
that have a weak derivativeWeak derivativeIn mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...
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 partsIntegration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
, 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.
In order to determine a finite-dimensional subspace
of 
consider a partitionPartition of an intervalIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the formIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form...
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 theoremTaylor's theoremIn calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...
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 normLp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
.
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 polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s for the subspace
-
