The finite element approximation

Chapter 7
The finite element approximation
”Pure mathematicians sometimes are satisfied with showing
that the non-existence of a solution implies a logical contradiction,
while engineers might consider a numerical result
as the only reasonable goal. Such one sided views seem to
reflect human limitations rather than objective values. In
itself mathematics is an indivisible organism uniting theoretical
contemplation and active application.”
Richard Courant (1888-1972)
The aim of this chapter is to introduce the basic theory of finite element methods. Nowadays, finite element
methods are widely used in almost every field of engineering analysis. The German mathematician
Richard Courant (1888-1972) shall probably be credited for formulating the essence of what is now called
a finite element [Courant, 1943]. The development of these methods became effective with the advent
of computers and is now recognized as one of the most powerful and versatile method for construction
approximations of the solutions of boundary-value problems. We will give here a brief overview of the
fundamental mathematical ideas that form the core of the method.
Contents
7.1 General principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 The one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 Triangular finite elements in higher dimensions . . . . . . . . . . . . . . . . . 120
7.4 The finite element method for the Stokes problem . . . . . . . . . . . . . . . 132
In Chapter 4 we have seen the principle of the variational approximation of elliptic problems. The main
idea of the finite element method is to replace the Hilbert space V in which the variational formulation
is posed by a finite dimensional subspace Vh. We will first briefly return to the internal variational
approximation principle and present finite elements in one dimension.
7.1 General principle
We consider the variational abstract problem introduced in Chapter 4. More precisely, given a Hilbert
space V , a bilinear continuous and V -elliptic form a(·, ·) defined on V × V and a continuous linear form
99

100 Chapter 7. The finite element approximation
l(·) defined on V ′ , we consider the variational formulation of a problem:
(P) : Find u ∈ V such that a(u, v) = l(v) , for all v ∈ V ,
and we already have stated that this problem has a unique solution using Lax-Milgram Theorem 4.1.
7.1.1 Internal approximation of a variational problem
The internal approximation of problem (P) consists then in replacing the Hilbert space V by a finite
dimensional subspace, denoted Vh, in which to find the solution uh. Here, h > 0 represents a parameter
related to the discretization of the domain (in space or time) and intended to vanish in the theoretical
results. We assume that for every v ∈ V , there exists an element rh(v) ∈ Vh such that
lim
h→0 �rh(v) − v� = 0 .
The bilinear form a(·, ·) and the linear form l(·) are defined on Vh × Vh and Vh, respectively and the
problem (P) is replaced by the following discrete problem:
(Ph) : Find uh ∈ Vh such that a(uh, vh) = l(vh) , for all vh ∈ Vh.
We introduce the keyword internal related to the approximation because we suppose here that Vh ⊂ V .
If N = dim Vh, we consider a basis (ϕi)1≤i≤N of Vh. The decomposition of uh in the basis of Vh,
uh = � N
i=1 uiϕi, leads to rewrite the problem (Ph) in the following form:
N�
uj a(ϕj, ϕi) = l(ϕi) , ∀1 ≤ i ≤ N . (7.1)
j=1
Introducing the stiffness matrix Ah = (aij) ∈ R N,N of coefficients aij = a(ϕj, ϕi), for all 1 ≤ i, j ≤ N,
the vector Uh = (ui)1≤i≤N and the vector Fh = (fi)1≤i≤N such that fi = l(ϕi)1≤i≤N allow us to conclude
that we have the equivalence:
uh solution of (Ph) ⇐⇒ AhUh = Fh .
Obviously, the V -ellipticity assumption on the bilinear form a implies the existence and the uniqueness
of the solution uh to the problem (Ph). However, this assumption is too strong for our purposes. As we
are considering a finite dimensional space Vh, it is sufficient to consider that:
(a(vh, vh) = 0) =⇒ (vh = 0) , ∀vh ∈ Vh ,
Since the bilinear form a is V -elliptic, then the matrix Ah is positive definite (cf. Proposition 4.2).
7.1.2 A priori error estimates
It is interesting to evaluate the error related to the replacement of V by the finite dimensional subspace
Vh. To this end, we assume that the problems (P) and (Ph) are both well-posed, in particular that the
bilinear form a(·, ·) is continuous (with a continuity constant M > 0) and that there exists a constant
αh > 0 such that:
∀vh ∈ Vh , a(vh, vh) ≥ αh�vh� 2 V ,
and we denote by u and uh the respective solutions of problems (P) and (Ph).
Proposition 7.1 Under the previous assumptions, we have the orthogonality identity:
∀vh ∈ Vh , a(u − uh, vh) = 0 .

7.1. General principle 101
Proof. Since Vh ⊂ V , we have trivially that a(u, vh) = l(vh) = a(uh, vh), for all vh ∈ Vh. �
If the bilinear form a(·, ·) is symmetric, continuous and V -elliptic on V × V of constant α, then it defines
an inner product and an energy norm associated to it as:
�u�e = (a(u, u)) 1/2 , ∀u ∈ V ,
equivalent to the norm of V :
√ α�u�V ≤ �u�e ≤ � �a� �u�V , ∀u ∈ V .
The approximate solution uh is thus the orthogonal projection for the inner product a(·, ·), of the solution
u on the subspace Vh.
A strong advantage of the internal variational approximation is that it provides an optimal estimate
of the error between the exact solution u of the problem (P) and the approximate solution uh of the
problem (Ph). The error �u − uh�V is comparable to the minimum of �u − vh�V when vh covers Vh. This
error estimate in norm � · �V is given by Céa’s lemma (cf. Chapter 4) that we recall here.
Lemma 7.1 (Céa) Under the previous hypothesis, we have the following error estimates.
(i) If the bilinear form is not V -elliptic, we have:
�u − uh�V
�
≤ 1 + �a�
�
inf �u − vh�V ,
αh vh∈Vh
where
a(vh, wh)
�a� = sup
.
vh,wh∈Vh �vh�V �wh�W
(ii) If the bilinear form a(·, ·) is continuous and V -elliptic with a coercivity constant α, then we have:
�u − uh�V ≤ M
α inf �u − vh�V .
vh∈Vh
(iii) If in addition the bilinear form a(·, ·) is symmetric the previous estimate becomes:
�
M
�u − uh�V ≤
α inf �u − vh�V .
vh∈Vh
Proof. Let consider wh ∈ Vh. From the orthogonality identity we deduce that
and taking into account the continuity of a, we write:
Similarly, we have
a(uh − wh, vh) = a(u − wh, vh) , ∀vh ∈ Vh .
a(uh − wh, vh) a(u − wh, vh)
αh�uh − wh�V ≤ sup
= sup
vh∈Vh �vh�V vh∈Vh �vh�V
a(u − uh, u − uh) = a(u − uh, u − vh) , ∀vh ∈ Vh ,
≤ �a��u − wh�V .
and we conclude with the V -ellipticity and the continuity of a. If the form a is symmetric, we can improve the
estimate. Since uh is the orthogonal projection of u onto Vh with respect to the inner product induced by a, we
have the Pythagorean relation:
a(u − uh, u − uh) = �u − uh� 2 e ≤ �u − wh� 2 e = a(u − wh, u − wh) , ∀wh ∈ Vh ,
and we deduce that for all wh ∈ Vh
α�u − uh� 2 V ≤ a(u − uh, u − uh) ≤ a(u − wh, u − wh) ≤ M�u − wh� 2 V .
and the results follows. �

102 Chapter 7. The finite element approximation
Corollary 7.1 Under the previous hypothesis, let (Vh)h denotes a set of finite dimensional subspaces of
V and let us assume that
∀v ∈ V , inf �v − vh�V
vh∈Vh
h→0
−→ 0 .
Then, if infh αh > 0, uh converges toward u in V .
The objective for the “ideal” approximation method is to define suitable approximation spaces Vh to
apply the Galerkin approach. To this end, we search for a compromise between the dimension N of Vh
(and thus the dimension of the matrix A) and the accuracy of the numerical solution uh. We shall also
consider spaces Vh that allow to compute easily the quantities a(ϕj, ϕi) and l(ϕi). Finally, specific spaces
Vh may result in sparse matrices A where the number of nonzero elements is small, or well-conditioned
matrices with a small condition number and thus easy to solve. In this spirit, the finite element method
tends to answer all these requirements. Before detailling the main concepts of the finite element methods,
we give a few words about Ritz and Petrov-Galerkin methods.
7.1.3 Ritz and Petrov-Galerkin methods
The Ritz method
We consider that the hypothesis of the Lax-Milgram theorem are satisfied and we recall that, if the
bilinear form a(·, ·) defined on V × V is symmetric, solving the problem:
(P) find u ∈ V, such that a(u, v) = l(v) , for all v ∈ V
is indeed equivalent to solving the problem:
( � 1
P) find u ∈ V, such that J(u) = inf J(v), where J(v) = a(v, v) − l(v).
v∈V 2
In the Ritz method, the space V is replaced by a finite dimensional subspace Vh ⊂ V such that
dim Vh = N and the approximate solution uh shall solve:
( � Ph) find uh ∈ Vh such that J(uh) = inf J(vh).
vh∈Vh
Theorem 4.2 ensures the existence of a unique solution to this minimization problem as a consequence of
Lax-Milgram theorem.
Since the dimension of the space Vh is N, there exists a basis (ϕj)1≤j≤N of Vh and every uh ∈ Vh can
N�
be decomposed as uh = uiϕj and we use the classical notation U = (u1, . . . , uN) t ∈ RN .
j=1
We consider the one-to-one mapping ξ : Vh → R N , uh ↦→ U and we pose J = J ◦ ξ −1 such that for
every uh ∈ Vh we have:
J (U) = J(uh) ,
or, when replacing J(uh) by its value:
J(uh) = 1
2 a
⎛
N� N�
⎝ ujϕj,
= 1
2
N�
j=1
i=1 j=1
j=1
ujϕj
N�
uiuja(ϕi, ϕj) −
⎞ ⎛
N�
⎠ − l ⎝
j=1
ujϕj
N�
uil(ϕi) .
This leads to a matrix formulation of the minimization functional J(u):
i=1
J(uh) = 1
2 U t Ah U − U t Fh = J (U) ,
⎞
⎠

7.1. General principle 103
where Ah = (aij) ∈ RN,N with aij = a(ϕi, ϕj) and Fh = (fi) ∈ RN is such that fi = l(ϕi). Hence, solving
the minimization problem ( � Ph) is equivalent to solving the following problem:
( � Ph,R) find U ∈ RN 1
such that J (U) = inf J (V ) , where J (V ) =
V ∈RN 2V t Ah V − V t Fh.
For obvious reasons, the stiffness matrix Ah is symmetric positive definite and thus the functional
J is quadratic on RN . This is sufficient to ensure the existence and uniqueness of U ∈ RN solving the
minimization problem ( � Ph,R). Furthermore, the solution U of the minimization problem ( � Ph,R) is also
the solution of the linear system AhU = Fh.
Remark 7.1 When the bilinear form a(·, ·) is symmetric, the Galerkin and Ritz methods are strictly
equivalent.
The Petrov-Galerkin method
The principles of Petrov-Galerkin and Galerkin methods are very similar in the sense that they both
will attempt to solve the problem (P). However, in the Petrov-Galerkin approach, we consider two
finite-dimensional approximation subspaces Vh and Wh in V such that
dim Vh = dim Wh = N .
The approximate solution uh is searched in the space Vh but the test functions in the variational formulation
are now the shape functions of Wh. For these reasons, Vh is called the approximation space and
Wh is the space of test functions. The problem to solve is now the following:
(Ph,P G) find uh ∈ Vh , such that a(uh, vh) = l(vh) , for all vh ∈ Wh.
Suppose (ϕj)1≤j≤N is a basis of Vh and (ψj)1≤j≤N a basis of Wh, then every uh ∈ Vh can be decomposed
N�
as uh = uiϕj and we can rewrite the problem as follows:
j=1
find uh ∈ Vh , such that
N�
uja(ϕj, ψi) = l(ψi) , i = 1, . . . , N,
j=1
And the linear system to solve is AhU = Fh, where Ah = (aij) ∈ R N,N with aij = a(ϕj, ψi) and
Fh = (fi) with fi = l(ψi), for all i = 1, . . . , N.
7.1.4 The finite element method
In the finite element method, the domain Ω is subdivided into a partition or a mesh Th, i.e., a (potentially
large) collection of geometrically simple elements, and the approximation space Vh is composed of
piecewise polynomial functions on each element K of the partition Th. We will see in Chapter 9 how to
construct a partition Th for a domain Ω of arbitrary geometric shape. The parameter h represents here
the grain of the discretization, i.e., the elementary size of the elements K in Th as defined by:
h = max diam(K) .
K∈Th
Typically, a basis of Vh will be composed of functions whose support is restricted on one or a few elements
of Th and the polynomials are usually of low degree. Hence, when h → 0 the space Vh will better and
better approximate the space V and the stiffness matrix A will be sparse, most of its coefficients being
zeros.

104 Chapter 7. The finite element approximation
Finite elements vs. finite differences or finite volumes
The principle of finite difference or finite volume discretization methods is very similar. Both approaches
consider a partition of the domain into a numerous small pieces, although none of them consider a
variational formulation of the problem at hand. For instance, let consider the homogeneous Dirichlet
boundary-value problem in two dimensions:
−∆u = f in Ω
u = 0 on ∂Ω
With the finite difference method, the domain Ω is covered by a regular uniform grid. At each internal
node xi,j = (ih, jh), we search a discrete value ui,j to approximate u(xi,j) and we assume for example
that the Laplacian operator is approximated using a 5 points scheme, thus leading to write:
4ui,j − ui,j+1 − ui,j−1 − ui+1,j − ui−1,j
h 2
= f(xi,j) .
In the finite volume method, the partition Th of the domain Ω is arbitrary and unknowns are associated
with each element K ∈ Th. Using Green’s identity, we can rewrite the previous equation as follows:
�
∂u
−
∂K ∂n =
�
f , ∀K ∈ Th
K
and we discretize the left-hand side term using a formula mixing the unknowns on K and on the neighboring
elements. Considering a square domain, we have:
Ki,j = [(i − 1/2)h, (i + 1/2)h] × [(j − 1/2)h, (j + 1/2)h] ,
and if ui,j is the approximation of u on Ki,j, the flux integrated on the interface Ki,j ∩ Ki+1,j is then
discretized by ui+1,l − uk,l. Repeating the procedure for the other fluxes and approximating the term
�
Ki,j f by h2 f(xi,j) yields the equation:
4ui,j − ui,j+1 − ui,j−1 − ui+1,j − ui−1,j = h 2 f(xi,j) .
The resulting linear system is very similar to that obtained using a finite difference scheme.
Actually, the vast majority of finite difference methods can be deduced from finite element methods if
the problem at hand has a variational formulation. It is less obvious for finite volume methods. Moreover,
we have strong theoretical mathematical tools to study finite element methods. In addition, the latter
have several intrinsic advantages:
1. the versatility of the formulation on arbitrarily complex geometries, and the possibility to locally
refine the partition Th to approximate solutions with singularities,
2. the boundary conditions are naturally taken into account in the space V in the variational formulation
and in its internal approximation Vh,
3. the general framework of the variational approximations is convenient for the error analysis.
Other variational methods have been developed, like spectral methods, that are especially adapted to
the approximation of smooth solutions but are limited to simple geometries and methods using wavelets
basis.

7.2. The one dimensional case 105
7.2 The one dimensional case
At first, we introduce the general principle of the Lagrange finite element method in one dimension of
space. Without loss of generality, we can restrict our study to the unit domain Ω =]0, 1[. To set the
ideas, we will also consider the following boundary-value problem:
Given f ∈ L 2 (Ω) and c ∈ L ∞ (Ω), find the function u solving:
� −u ′′ (x) + c(x)u(x) = f(x) , x ∈]0, 1[
u(0) = u(1) = 0 .
Here, a mesh is simply a set of points (xj)0≤j≤N+1 or intervals Kj = [xj, xj+1] such that 0 = x0 <
x1 < · · · < xN+1 = 1. The mesh is said to be uniform if the points (xj) are equidistributed along the
segment [0, 1], i.e. such that xj = jh, with h = 1/(N + 1), 0 ≤ j ≤ N + 1. More generally, we denote by
h = max |xj+1 − xj| the size parameter.
7.2.1 Lagrange P1 elements
The finite element methods for Lagrange P1 elements involves the space of globally continuous affine
functions on each interval:
and the subspace of V 1 h :
V 1 h = {vh ∈ C 0 ([0, 1]) , vh|Kj ∈ P1 , 0 ≤ j ≤ N} ,
V 1
0,h = {vh ∈ V 1 h , such that vh(0) = vh(1) = 0} ,
More generally, Pk denotes the vector space of polynomials in one variable and of degree less than or
equal to k:
⎧
⎨
k�
Pk = p(x) = αjx
⎩ j ⎫
⎬
αj ∈ R
⎭ .
j=0
The finite element method consists in applying the internal variational approximation approach to the
spaces V 1 h and V 1
0,h . In this context, the functions of V 1 h can be represented using very simple shape
functions.
Lemma 7.2 The space V 1 h is a subspace of H1 (Ω) of dimension N + 2. Every function vh of V 1 h is
uniquely determined by its values at the mesh vertices (xj)0≤j≤N+1:
vh(x) =
N+1 �
j=0
vh(xj)ϕj(x) , ∀x ∈ [0, 1] ,
where (ϕj)0≤j≤N+1 is the basis of the shape functions ϕj with compact support in each interval [xj−1, xj+1]
defined as:
⎧
⎪⎨
x − xj−1
ϕj(x) =
h
⎪⎩
xj+1 − x
h
x ∈ [xj−1, xj]
x ∈ [xj, xj+1]
(7.2)
such that ϕj(xi) = δij . (7.3)

106 Chapter 7. The finite element approximation
1
x0
ϕ0
xj−1
ϕj
xj
xj+1
ϕN+1
xN+1
Figure 7.1: Global shape functions for the space V 1 h .
Proof. We know that piecewise C 1 continuous functions belong to the space H 1 (Ω). Hence, Vh is a subspace of
H 1 (Ω). Moreover, since we have ϕj(xi) = δij, where δij is the Kronecker symbol, the result follows. �
Remark 7.2 Notice that the functions (ϕj)0≤j≤N+1 can be expressed using only two functions:
ω0(x) = 1 − x , ω1(x) = x .
The basis functions are defined as the composition of a shape function of a reference finite element (i.e.
that depends only of the polynomial approximation) and of an affine transformation (depending only on
the discretization) as, for all 0 ≤ j ≤ N:
⎧ � �
x − xj−1
⎪⎨ ω1
, x ∈ [xj−1, xj]
xj − xj−1
ϕj(x) = � �
⎪⎩
x − xj
, x ∈ [xj, xj+1]
ω0
xj+1 − xj
and ϕN+1(x) = ω1 ((x − xN)/h). This will be useful to compute the coefficients of the matrix in the linear
system to solve.
Corollary 7.2 The space V 1
uniquely determined by its values at the mesh vertices (xj)1≤j≤N:
0,h is a subspace of H1 0 (Ω) of dimension N and every function vh of V 1
vh(x) =
N�
vh(xj)ϕj(x) , ∀x ∈ [0, 1] ,
j=1
Remark 7.3 Notice that functions vh ∈ V 1 h are not twice differentiable on Ω and thus it is meaningless
to attempt solving the problem (7.2) as the second derivative of any vh ∈ V 1 h is a sum of Dirac masses
at the mesh vertices. However, it is meaningful to solve a variational formulation of this problem with
functions vh ∈ V 1 h since only the first derivatives are involved.
The variational formulation of problem (7.2) consists in finding u ∈ H 1 0
�
u
Ω
′ (x)v ′ (x) dx +
�
Ω
�
c(x)u(x)v(x) dx =
(Ω), such that:
0,h is
f(x)v(x) dx , ∀v ∈ H
Ω
1 0 (Ω) , (7.4)
and the variational formulation of the internal approximation consists in finding uh ∈ V0,h, such that:
�
�
�
c(x)uh(x)vh(x) dx = f(x)vh(x) dx , ∀vh ∈ V0,h . (7.5)
Ω
u ′ h (x)v′ h (x) dx +
Ω
Ω

7.2. The one dimensional case 107
Introducing the notation uh(xj)1≤j≤N for the approximate value of the exact solution at the mesh vertex
xj, leads to the approximate problem:
find uh(x1), . . . , uh(xN) such that for all i = 1, . . . , N
N�
��
ϕ
Ω
′ j(x)ϕ ′ �
i(x) dx +
j=1
Ω
� �
c(x)ϕj(x)ϕi(x) dx uh(xj) = f(x)ϕi(x) dx .
Ω
And as expected, this formulation is equivalent to solving in R N the linear system:
where Uh = (uh(xj))1≤j≤N, Fh = ��
Coefficients of the matrix Ah
AhUh = Fh
Ω f(x)ϕi(x) dx �
��
Ah = ϕ
Ω
′ j(x)ϕ ′ �
i(x) dx +
1≤i≤N and the matrix Ah is defined as:
Ω
�
c(x)ϕj(x)ϕi(x) dx
1≤i,j≤N
Actually, the matrix Ah appears as the sum of the stiffness matrix Kh defined by its coefficients
(kij)1≤i,j≤N:
�
� xk+1
kij =
Ω
ϕ ′ j(x)ϕ ′ i(x) dx =
and the mass matrix Mh defined by its coefficients (mij)1≤i,j≤N:
�
mij =
Ω
c(x)ϕj(x)ϕi(x) dx =
N�
k=0
N�
k=0
xk
� xk+1
xk
ϕ ′ j(x)ϕ ′ i(x) dx ,
c(x)ϕj(x)ϕi(x) dx .
Since the shape functions ϕj have a small support, most of the coefficients in Ah are zeros. More precisely,
for a given index i, there is only three consecutive values of j such that the coefficient aij is potentially
not equal to zero. The structure of the matrice is then easy to deduce: Ah is a tridiagonal matrix. The
coefficients of Ah are thus given by:
ajj =
ajj−1 =
ajj+1 =
� xj+1
xj−1
� xj
xj−1
� xj+1
xj
(ϕ ′ j(x)) 2 � xj+1
dx +
ϕ ′ j(x)ϕ ′ j−1(x) dx +
xj−1
� xj
ϕ ′ j(x)ϕ ′ j+1(x) dx +
c(x)(ϕj(x)) 2 dx
xj−1
� xj+1
xj
c(x)ϕj(x)ϕj−1(x) dx
c(x)ϕj(x)ϕj+1(x) dx
For the sake of simplicity, we consider here the function c as being constant, c(x) = c0 for all x ∈ Ω.
Hence, we write:
� � � � � �
xj
xj x − xj−1 x − xj−1
mjj = c0 ϕj(x)ϕj−1(x) dx = c0 ω1
ω0
h
h
xj−1
= c0h
xj−1
� 1
0
ω1(y)ω0(y) dy = c0h
where we posed y = (x − xj−1)/h. Finally, we find the coefficients of Ah:
ajj−1 = − 1
h
h
+ c0
6
ajj = 2 2h
+ c0
h 3
� 1
and ajj+1 = − 1
h
0
.
h
(1 − y)y dy = c0
6 ,
+ c0
h
6 ,

108 Chapter 7. The finite element approximation
Remark 7.4 Instead of regarding the node contributions, we could have analyzed the elements. Consider
the element Kj = [xj, xj+1]; on this element there is only two non-zero shape functions:
ϕj|Kj = xj+1 − x
xj+1 − xj
ϕ ′ j|Kj =
−1
xj+1 − xj
= xj+1 − x
h
= −1
h
ϕj+1|Kj
ϕ ′ j+1|Kj =
= x − xj
xj+1 − xj
1
xj+1 − xj
= x − xj
= 1
h
h
Then, we can arrange the elementary contributions of the element Kj to the stiffness matrix and to the
mass matrix as 2 × 2 symmetric matrices EKj and EMj:
�
�
with
k j
11 =
=
EKj =
� xj+1
(ϕ ′ j(x)) 2 dx ,
xj
� xj+1
xj
1 1
dx =
h2 h
�
k j
11 kj 12
k j
21 kj 22
k j
12 = kj 21 =
� xj+1
=
xj
� xj+1
xj
and EMj =
ϕ ′ j(x)ϕ ′ j+1(x) dx ,
− 1 1
dx = −
h2 h
�
m j
11 mj12
m j
21 mj22
k j
22 =
m j
11 =
� xj+1
c(x)(ϕj(x))
xj
2 dx , m j
12 = mj21
=
� xj+1
c(x)ϕj(x)ϕj+1(x) dx , m
xj
j
22 =
and thus to conclude that:
EKj = 1
h
�
1
�
−1
−1 1
h
and EMj = c0
6
=
� �
2 1
1 2
� xj+1
(ϕ ′ j+1(x)) 2 dx
xj
� xj+1
xj
� xj+1
xj
1 1
dx =
h2 h
c(x)(ϕj+1(x)) 2 dx
We will see that this point of view is more practical when dealing with the matrix assembly, especially in
higher dimensions.
Matrix assembly
The assembly of the matrix Ah is easy and is obtained algorithmically using a loop over all mesh elements
Kj and adding their contributions to the right coefficients of the global system. Assuming a(i, j) denotes
the coefficients aij of Ah, a pseudo-code to perform this task would be:
for k=1, N+1 do // loop over all elements
for i=1,2 do // local loop
for j=1,2 do
ig = k+i-2 // global indices
jg = k+j-2
A(ig,jg) = A(ig,jg) + a(i,j)
end loop j
end loop i
end loop k
The numerical resolution of the linear system is by far the most computationally expensive part of the
method. We refer the reader to Chapter 5 for more details about the direct and indirect techniques to
solve this system.

7.2. The one dimensional case 109
Coefficients of the right-hand side Fh
Each component fi of the vector Fh ∈ R N is obtained as:
fi =
N�
k=0
� xk+1
xk
f(x)ϕi(x) dx .
Usually, the function f is not known analytically. Hence, we decompose f in the basis of the shape
functions (ϕj)1≤j≤N:
f(x) =
N−1 �
j=1
fjϕj(x) dx
and the problem is reduced to the evaluation of the integrals:
We use for instance the trapeze formula:
� xk+1
xk
� xk+1
xk
ϕj(x)ϕi(x), dx .
θ(x) dx = xk+1 − xk
(θ(xk+1) + θ(xk)) ,
2
that gives the exact result for polynomial of degree one and leads here fj = hf(xj); or the Simpson
formula: � xk+1
θ(x) dx = xk+1 − xk
(θ(xk+1) + 4θ(x
6
k+1/2) + θ(xk)) ,
xk
that gives the exact result for polynomials of degree lesser than or equal to 3, and leads here to
Neumann boundary-value problem
fj = h
6 (f(xj) + 4f(x j+1/2) + f(xj+1)) .
The finite element method can be applied to solve the Neumann boundary-value problem:
Given f ∈ L2 (Ω), c ∈ L∞ (Ω) such that c(x) ≥ c0 > 0 almost everywhere in Ω and α, β ∈ R, find the
function u solving: �
′′
−u (x) + c(x)u(x) = f(x) , x ∈ Ω
u ′ (0) = α u ′ (7.6)
(1) = β .
in a very similar manner. Recall that this problem has a unique solution u ∈ H1 (Ω) (cf. Chapter 4).
The variational formulation of the internal approximation consists here in finding uh ∈ V 1 h such that
�
u ′ h (x)v′ �
�
h (x) dx + c(x)uh(x)vh(x) dx = f(x)vh(x) dx − αvh(0) + βvh(1) , ∀vh ∈ V 1 h (Ω) . (7.7)
Ω
Ω
The variational formulation consists in solving in R N+2 the linear system:
Ω
AhUh = Fh
with Uh = (uh(xj))0≤j≤N+1 and the stiffness matrix Ah is defined as:
��
Ah = ϕ
Ω
′ j(x)ϕ ′ �
i(x) dx +
Ω
�
c(x)ϕj(x)ϕi(x)) dx
0≤i,j≤N+1
.

110 Chapter 7. The finite element approximation
and Fh = (fj)0≤j≤N+1 such that:
�
fj = f(x)ϕj dx 1 ≤ j ≤ N
�Ω
f0 = f(x)ϕ0(x) dx − α
�Ω
fN+1 = f(x)ϕN+1(x) dx + β .
7.2.2 Convergence of the Lagrange P1 finite element method
Ω
Definition 7.1 (Interpolation) The linear mapping Πh : H 1 (Ω) → V 1 h defined for every v ∈ H1 (Ω)
as:
(Πhv)(x) =
N+1 �
j=0
v(xj)ϕj(x) , ∀x ∈ [0, 1] ,
is called P1 interpolation operator. Furthermore, for every v ∈ H 1 (Ω), the interpolation operator is such
that:
lim
h→0 �v − Πhv� H 1 (Ω) = 0 .
The P1 interpolate of a function v is the unique piecewise affine function that coincide with v at the mesh
vertices xj. The convergence of the finite element method is related to a series of results that we give
here.
Suppose that the function v is sufficiently smooth, i.e. v ∈ H 2 (Ω). Since the derivative of the affine
functions in Vh is constant on the intervals Kj = [xj, xj+1], we have then:
� xj+1
(Πhv) ′ (x) = v(xj+1) − v(xj)
h
= 1
h
xj
v ′ (t) dt , ∀x ∈ [xj, xj+1] .
Since we assumed v ∈ H 2 (Ω) then v ′ ∈ H 1 (Ω) and thus v is a continuous function. Using Rolle’s theorem,
we deduce that there exists a point θj ∈ [xj, xj+1] such that:
v ′ (θj) = 1
h
� xj+1
v ′ (t) dt = (Πhv) ′ (x) , ∀x ∈ [xj, xj+1] .
xj
We will search for an estimate on �v − Πhv� H 1 (Ω). To this end, we write:
�v − Πhv� 2 H 1 (Ω) =
however, for all t ∈ [xj, xj+1] we have:
� 1
|v
0
′ − Π ′ hv|2 N−1 �
� xj+1
= |v
xj j=1
′ (t) − v ′ (θj)| 2 dt , (7.8)
v ′ (t) − v ′ � t
(θj) =
hence, using Cauchy-Schwarz’s identity, we write:
|v ′ (t) − v ′ (θj)| 2 ≤
≤
� t
θj
� xj+1
xj
θj
v ′′ (t) dt ,
|v ′′ (t)| 2 dt |t − θj|
|v ′′ (t)| 2 dt |t − θj| .

7.2. The one dimensional case 111
By integrating on the interval [xj, xj+1] yields:
� xj+1
|v ′ (t) − v ′ (θj)| 2 � xj+1
dt ≤
xj
|t − θj| dt
� � xj+1
|v ′′ (t)| 2 dt
xj
xj
≤ (xj+1 − xj) 2 � xj+1
|v ′′ (t)| 2 dt
≤ h2
2
And going back to equation (7.8), we have now:
�v − Πhv� H 1 (Ω) ≤ h2
2
2
xj
� xj+1
|v ′′ (t)| 2 dt .
xj
N−1 �
j=1
� xj+1
xj
≤ h2
2 �v′′ � 2 L 2 (Ω) .
|v ′′ (t)| dt
This leads to enounce the following result, that we already partly proved.
Lemma 7.3 (Interpolation error) If v ∈ H 2 (Ω) then, there exists two constant C1 and C2 independent
of h such that:
�v − Πhv� H 1 (Ω) ≤ C1 h 2 �v ′′ � L 2 (Ω) and �v ′ − (Πhv) ′ � L 2 (Ω) ≤ C2 h �v ′′ � L 2 (Ω) .
And we can establish the convergence of the finite element method for the Dirichlet boundary-value
problem as follows.
Theorem 7.1 (Convergence) Suppose u ∈ H 1 0 (Ω) and uh ∈ V0,h are the solutions of (7.2) and (7.5),
respectively. Then, the Lagrange P1 finite element method converges, i.e. we have:
lim
h→0 �u − uh�H 1 (Ω) = 0 .
Furthermore, if u ∈ H 2 (Ω) then, there exists a constant C independent of h such that:
�u − uh� H 1 (Ω) ≤ Ch�f� L 2 (Ω) .
Proof. Here, since the bilinear form a(·, ·) is V0,h-elliptic, we can consider the ellipticity constant α = 1:
� 1
a(u, u) =
and regarding the continuity of a(·, ·) we write:
0
u ′2 (x) + cu 2 (x) dx ≥
|a(u, v)| ≤
� 1
0
u ′2 (x) dx = �u� 2
H1 0 (Ω) ,
�
�u ′ � 2 L2 (Ω) + c1�u� 2 L2 �1/2 (Ω)
≤ (1 + c1)�u� H 1 0 (Ω) �v� H 1 0 (Ω) ,
�

112 Chapter 7. The finite element approximation
where we assumed that 0 ≤ c1 = sup x∈[0,1] c(x) ≤ +∞. Thus, the continuity constant M is taken here as 1 + c1.
Using Céa’s lemma 7.1, we can easily conclude that:
�u − uh�H 1
0 (Ω) ≤ √ 1 + c1 inf �u − vh�H 1
vh∈V0,h
0 (Ω) . (7.9)
If we assume v ∈ H 2 (Ω), we can write, according to the previous lemma:
Moreover, since Πhu ∈ V0,h, we have also:
�u − Πhu� 2
H1 h2
0 (Ω) ≤
2 �u′′ � 2 L2 (Ω)
≤ h2
2 �u�2 h2
H2 ≤ C
2 �f�2L 2 (Ω) .
inf �u − vh�H 1
vh∈V0,h
0 (Ω) ≤ �u − Πhu�H 1
0 (Ω) .
and using the inequality (7.9), we can conclude. �
Lemma 7.4 There exists a constant C independent of h such that for all v ∈ H 1 (Ω):
�Πhv� H 1 (Ω) ≤ C�v� H 1 (Ω) , and �v − Πhv� L 2 (Ω) ≤ Ch�v ′ � L 2 (Ω) .
Furthemore, for all v ∈ H 1 (Ω), we have:
Proof. Given v ∈ H 1 (Ω), we have:
lim
h→0 �v′ − (Πhv) ′ �L2 (Ω) = 0 .
�Πhv�L2 (Ω) ≤ sup |Πhv(x)| ≤ sup |v(x)| ≤ C�v�H 1 (Ω) .
x∈Ω
x∈Ω
Moreover, since Πhv is an affine function, we have by Cauchy-Schwarz’s identity:
� xj+1
|(Πhv) ′ (t)| 2 dt = (v(xj+1) − v(xj)) 2
=
h
1
�� xj+1
v
h
′ �2 � xj+1
(t) dt ≤
xj
and we obtain the first identity by summation over j. Similarly, we write:
� xj+1
|v(x) − Πhv(x)| ≤ 2 |v ′ (t)|dt .
xj
xj
xj
|v ′ (t)| 2 dt .
We obtain the second identity by using Cauchy-Schwarz, by integrating with respect to x and then by summation
over j.
Since C ∞ (Ω) is dense in H 1 (Ω), for every v ∈ H 1 (Ω) there exists w ∈ C ∞ (Ω) such that
�v ′ − w ′ � L 2 (Ω) ≤ ε , for ε > 0 .
Since Πh is a linear mapping verifying the first identity, we have then:
�(P ihv) ′ − (Πhw) ′ � L 2 (Ω) ≤ C�v ′ − w ′ � L 2 (Ω) ≤ Cε .
From Lemma 7.3, we deduce that, for h sufficiently small:
We can the write, by adding the last identities:
�w ′ − (Πhw) ′ � L 2 (Ω) ≤ ε .
�v ′ − (Πhv) ′ � L 2 (Ω) ≤ �v ′ − w ′ � L 2 (Ω) + �w ′ − (Πhw) ′ � L 2 (Ω) + �(Πhv) ′ − (Πhw) ′ � L 2 (Ω) ≤ Cε ,
and the result follows. �

7.2. The one dimensional case 113
7.2.3 Lagrange P2 elements
Before introducing the Lagrange P2 finite element method, we like to describe the advantage of considering
higher-order polynomials on an example taken from [ ˇ Solin, 2005].
Motivation for high-order elements
Consider the simple homogeneous Poisson boundary-value problem in one dimension of space:
�
−u ′′ (x) = f(x) ,
u(0) = u(1) = 0 .
in Ω =] − 1, 1[ ,
, with f(x) = π2
4 cos
�
πx
�
.
2
The exact solution to this problem has the form:
u(x) = cos
�
πx
�
.
2
The well-known variational formulation of this problem consists in: finding u ∈ H 1 0
�
u
Ω
′ (x)v ′ (x) dx =
�
f(x)v(x) dx , ∀v ∈ H
Ω
1 0 (Ω)
(Ω) such that:
Suppose the domain is decomposed into two intervals [−1, 0] and [0, 1] and let consider the finite element
space V 1
0,h generated by a single piecewise affine function vh defined as:
vh(x) =
� x + 1 , x ∈ [−1, 0]
1 − x , x ∈ [0, 1]
The exact solution u and the approximate solution uh are given Figure 7.2, left. The approximation error
in H1 seminorm is:
��
�1/2 |u − uh|1,2 =
≈ 0.683667 .
|u
Ω
′ (x) − u ′ h (x)|2 dx
On the other hand, assume a single quadratic element covers the domain [−1, 1]. A basis of the finite
element space V 2
0,h is composed of the function vh(x) = 1−x 2 . The exact solution u and the approximate
solution uh are given Figure 7.2, right. The approximation error is then:
0.8
0.6
0.4
0.2
1 u
1-x
x+1
0
-1 -0.5 0 0.5 1
|u − uh(x)|1,2 ≈ 0.20275 .
0.8
0.6
0.4
0.2
1 u
1-x^2
0
-1 -0.5 0 0.5 1
Figure 7.2: .Exact solution u with piecewise affine approximation uh (left-hand side) and with quadratic
approximation uh (right-hand side).

114 Chapter 7. The finite element approximation
This is a clear indication that high-order finite elements are better to approximate smooth functions.
Conversely, less regular functions can be approximated accurately using lower degree finite elements.
This will be emphasized by Theorem 7.2.
Lagrange P2 elements
We return to problem (7.2) and we consider a set of points (xj)0≤j≤N+1 or intervals Kj = [xj, xj+1]
forming a uniform mesh of Ω. The finite element method for Lagrange P2 elements involves the discrete
space:
V 2 h = {vh ∈ C 0 ([0, 1]) , vh|Kj ∈ P2 , 0 ≤ j ≤ N} ,
and its subspace:
V 2
0,h = {vh ∈ V 2 h , such that vh(0) = vh(1) = 0} .
These spaces are composed of continuous, piecewise parabolic functions (polynomials of degree less than
or equal to 2). The P2 finite element method consists in applying the internal variational approximation
approach to these spaces.
Lemma 7.5 The space V 2 h is a subspace of H1 (Ω) of dimension 2N + 3. Every function vh ∈ V 2 h is
uniquely defined by its values at the mesh vertices (xj)0≤j≤N+1 and at the midpoints (x j+1/2)0≤j≤N =
(xj + h
2 )0≤j≤N:
vh(x) =
N+1 �
j=0
vh(xj)ϕj(x) +
N�
vh(xj+1/2) ϕj+1/2(x) , ∀x ∈ [0, 1] .
where (ϕj)0≤j≤N+1 is the basis of the shape functions ϕj defined as:
with
j=0
� �
� �
x − xj
x − xj+1/2
ϕj(x) = φ , 0 ≤ j ≤ N + 1 and ϕ
h
j+1/2(x) = ψ
, 0 ≤ j ≤ N ,
h
⎧
⎪⎨ (1 + x)(1 + 2x) − 1 ≤ x ≤ 0
φ(x) = (1 − x)(1 − 2x)
⎪⎩
0
0 ≤ x ≤ 1
|x| > 1 ,
Remark 7.5 Notice that we have:
ϕi(xj) = δij
and ψ(x) =
ϕi(x j+1/2) = 0
ϕ i+1/2(xj) = 0 ϕ i+1/2(x j+1/2) = δij
� 1 − 4x 2
|x| ≤ 1/2
0 |x| > 1/2
Corollary 7.3 The space V 2
is uniquely defined by its values at the mesh vertices (xj)1≤j≤N and at the midpoints (xj+1/2)0≤j≤N: vh(x) =
0,h is a subspace of H1 0 (Ω) of dimension 2N + 1 and every function vh ∈ V 2
0,h
N�
vh(xj)ϕj(x) +
j=1
N�
vh(xj+1/2) ϕj+1/2(x) , ∀x ∈ [0, 1] .
j=0

7.2. The one dimensional case 115
1
xj−1
x j−1/2
ϕj
xj
ϕ j+1/2
x j+1/2
xj+1
Figure 7.3: Global shape functions for the space V 2 h .
The variational formulation of the internal approximation of the Dirichlet boundary-value problem (7.2)
, such that:
consists now in finding uh ∈ V 2
0,h
�
u
Ω
′ h (x)v′ �
�
h (x) dx + c(x)uh(x)vh(x) dx =
Ω
f(x)vh(x) dx ,
Ω
∀vh ∈ V 2
0,h . (7.10)
Here, it is convenient to introduce the notation (xk/2)1≤k≤2N+1 for the mesh points and (ϕk/2)1≤k≤2N+1 for the basis of V 2
0,h . Using these notations, we have:
uh(x) =
2N+1 �
k=1
This formulation leads to solve in R 2N+1 a linear system:
uh(x k/2)ϕ k/2(x) .
AhUh = Fh ,
where Uh = (uh(x k/2))1≤k≤2N+1 and it is easy to see that the matrix Ah of the linear system to solve is
now defined as:
��
Ah = ϕ
Ω
′ k/2 (x)ϕ′ �
l/2 (x) dx +
and the right-hand side term becomes:
��
�
Fh = f(x)ϕk/2(x) dx
Ω
Ω
�
c(x)ϕk/2(x)ϕl/2(x) dx
,
1≤k,l≤2N+1
1≤k≤2N+1
Since the shape functions ϕj have a small support, the matrix Ah is mostly composed of zeros. However,
the main difference with the Lagrange P1 finite element method, the matrix Ah is no longer a tridiagonal
matrix.
Coefficients of Ah
The coefficients of the matrix Ah can be coputed more easily by considering the following change of
variables, for t ∈ [−1, 1]:
x = xj+1 + xj+2 − xj
t = xj+1 +
2
h
2 t , ∀x ∈ [xj, xj+2] , 0 ≤ j ≤ 2N − 1 .
.

116 Chapter 7. The finite element approximation
Hence, the shape functions can be reduced to only three basic shape functions (Figure 7.4):
ω−1(t) =
and their respective derivatives:
t(t − 1)
2
dω−1 2t − 1
(t) =
dt 2
ω0(t) = −(t − 1)(t + 1) ω1(t) =
dω0
(t) = −2t
dt
t(t + 1)
2
dω1 2t + 1
(t) = .
dt 2
This approach consists in considering all computations on an interval Kj = [xj, xj+2] on the reference
interval [−1, 1]. Thus, we have:
dϕj
dx
= dωk
dt
where ωk ∈ [−1, 1]. In this case, the elementary contributions of the element Kj to the stiffness matrix
and to the mass matrix are given by the 3 × 3 matrices EKj and EMj:
dt
dx ,
EKj = 1
⎛
⎞
⎛
⎞
7 −8 1
4 2 −1
⎝−8
16 −8⎠
h
EMj = c0 ⎝ 2 16 2 ⎠ .
3h
30
1 −8 7
−1 2 4
1
0.8
0.6
0.4
0.2
0
-1 -0.5 0 0.5 1
Figure 7.4: The three quadratic Lagrange P2 shape functions on the reference interval [−1, 1].
Matrix assembly
for k=1, N do // loop over all elements
for i=1,3 do // local loop
for j=1,3 do
ig = 2*k+i-3 // global indices
jg = 2*k+j-3
A(ig,jg) = A(ig,jg) + a(i,j)
end loop j
end loop i
end loop k
,

7.2. The one dimensional case 117
Coefficients of the right-hand side Fh
Usually, the function f is only known by its values at the mesh points (xj)0≤j≤2N and thus, we use the
decomposition of f in the basis of shape functions (ϕj)0≤j≤2N:
f(x) =
2N�
j=0
f(xj)ϕj(x) dx .
Each component fi of the right-hand side vector is obtained as:
fi =
N�
k=1
� x2k
x2k−2
Using the previous decomposition of f, we obtain:
fi =
2N�
fj
j=0
� N�
k=1
� x2k
x2k−2
and the problem is reduced to computing the inmtegrals:
� x2k
x2k−2
f(x)ϕi(x) dx .
ϕj(x)ϕi(x) dx
ϕj(x)ϕi(x) dx, .
It is easy to see that we obtain expressions very similar to that of the mass matrix. More precisely, the
element Kj = [xi, xi+2] will contribute to only three components of indices i, i+! and i + 2 as:
where f k i
⎛
⎝
f k i
f k i+1
f k i+2
⎞
⎠ = h
30
⎛
4 2
⎞ ⎛
−1
⎝ 2 16 2 ⎠ ⎝
−1 2 4
fi
fi+1
fi+2
denotes the contribution of element k to the component i.
7.2.4 Convergence of the Lagrange P2 finite element method
We rely on Céa’s lemma that provides an estimate of the error:
�u − uh� H 1 (Ω) ≤
�
⎞
,
⎠ ,
�
M
α inf
vh∈V 1
�u − vh�H 1 (Ω) ,
0,h
for a continuous and V -elliptic bilinear form a(·, ·) defined on V 1 h × V 1 h . Suppose now that f ∈ H1 (Ω),
then u ∈ H 3 (Ω) and if the function c is sufficiently smooth then:
�u� H 3 (Ω) ≤ C �f� H 1 (Ω) .
In order to find an upper bound on the right-hand side of the previous estimate, we introduce a mapping
wh ∈ V 2
0,h such that for all 1 ≤ i ≤ N, wh(xi) = u(xi) and such that wh| [xi,xi+1] is a polynomial of degree
two or less. To this end, on each interval [xi, xi+1], we consider the polynomial functions:
wi,1(x) = αi
2 (x − xi) 2 , wi,2(x) = wi,1(x) + βi(x − xi) ,

118 Chapter 7. The finite element approximation
where
By definition, we have:
� xi+1
xi
(u − wi,1) ′′ (t) dt = 0 ,
αi = 1
� xi+1
u
h xi
′′ (t) dt , βi = 1
h
� xi+1
xi
� xi+1
xi
(u − wi,2) ′′ (t) dt = 0 ,
Hence, from this relation, we deduce that, for every 0 ≤ i ≤ N:
u(xi) − wi,2(xi) = u(xi+1) − wi,2(xi+1) .
This allow us to define the polynomial function wh on [0, 1] as follows:
(u − wi,1) ′ (t) dt
� xi+1
wh(x) = wi,2(x) + (u(xi) − wi,2(xi)) , ∀ 0 ≤ i ≤ N ,
�
M
α inf
vh∈V 1
�u − vh�H 1 (Ω) ≤
0,h
xi
(u − wi,2) ′ (t) dt = 0 .
and the previous relations show that wh is defined and continuous on [0, 1], that wh(xi) = u(xi) for all
0 ≤ i ≤ N and that wh is a polynomial of degree 2 on each [xi, xi+1]. We conclude easily that:
�
M
�u − uh�H 1 (Ω) ≤
α �u − wh�H 1 (Ω) .
Introducing the notation rhu = u − wh, we can see from the previous identities that rhu ∈ H 3 (Ω) and
rhu| [xi,xi+1] ∈ H 1 0 ([xi, xi+1]). Furthermore, we have:
� xi+1
xi
(rhu) ′′ (t) dt = 0 , and
� xi+1
xi
(rhu) ′ (t) dt = 0 .
To achieve the estimate of Rhu, we introduce a result known as Poincaré-Wirtinger inequality.
Lemma 7.6 (Poincaré-Wirtinger inequality) Given a bounded interval [a, b] of R, we pose
Then, we have:
� b
a
W [a,b] =
u 2 (t) dt ≤
�
u ∈ H 1 ([a, b]) ,
b − a
2
� b
a
�
u(t) dt = 0 .
� b
a
u ′ (t) dt , ∀ u ∈ W [a,b] .
Using this result and the previous identities, we deduce that r ′′
h u ∈ W [xj,xi+1], for all 0 ≤ i ≤ N and thus:
� xi+1
Similarly, we can deduce that:
� xi+1
xi
xi
|(rhu) ′′ (t)| 2 dt ≤ h2
2
≤ h2
2
|(rhu) ′ (t)| 2 dt ≤ h2
2
≤ h4
4
� xi+1
|(rhu) ′′′ (t)| 2 dt
xi
� xi+1
|u ′′′ (t)| 2 dt
xi
� xi+1
|(rhu) ′′ (t)| 2 dt
xi
� xi+1
|u ′′′ (t)| 2 dt
xi
.
.

7.2. The one dimensional case 119
Adding these last two results yields:
�rhu� 2
H1 � xi+1
0 (Ω) = |(rhu)
xi
′ (t)| dt ≤ h4
4 �u′′′ �L2 (Ω) .
Finally, we can enounce the convergence result as follows.
Theorem 7.2 (Convergence) Suppose u ∈ H1 0 (Ω) and uh ∈ V 2
0,h
are the solutions of (7.2) and (7.10),
respectively. Then, the Lagrange P2 finite element method converges, i.e. we have:
lim
h→0 �u − uh�H 1 (Ω) = 0 .
Furthermore, if u ∈ H 3 (Ω) (i.e. f ∈ H 1 (Ω)), then there exists a constant C independent of h such that:
�u − uh� H 1 (Ω) ≤ C h 2 �u ′′′ � L 2 (Ω) .
Remark 7.6 The convergence rate of the P2 finite element method is better than with the P1 finite
element method. However, the data f must be sufficiently smooth, here f ∈ H 1 .
7.2.5 Lagrange Pk elements
This section generalizes the concepts introduced in the previous sections to the interpolation of continuous
and polynomials functions of degree k ≥ 1. We consider a set of points (xj)0≤j≤N+1 or intervals Kj =
[xj, xj+1] forming a uniform mesh of Ω =]0, 1[.
Lagrange finite element spaces
For a given integer k ≥ 1, we define the space of globally continuous functions on [0, 1] whose restriction
on each interval Kj = [xj, xj+1] is a polynomial of degree k:
and the subspace of V k h :
V k h = {vh ∈ C 0 ([0, 1]) , vh|Kj ∈ Pk , 0 ≤ j ≤ N} ,
V k
0,h = {vh ∈ V k h , such that vh(0) = vh(1) = 0} .
In each interval Kj, a function of V k h is uniquely determined by its values at k + 1 distinct points along
the segment. Hence, on each interval Kj, we introduce a set of nodes:
and yN+1,0 = xN+1.
yj,l = xj + l
k (xj+1 − xj) = xj + l
k hj , 0 ≤ l ≤ k − 1 ,
Lemma 7.7 The space V k h is a subspace of H1 (Ω) of dimension k(N + 1) + 1. Every function vh of V k h
is uniquely determined by its values at the mesh nodes (yj,l)0≤j≤N,0≤l≤k−1 and yN+1,0. Furthermore, the
shape functions are such that:
ϕj,l(yj ′ ,l ′) = δjj ′δll ′ .
The space V k
0,h is a subspace of H1 0 (Ω) of dimension k(N + 1) − 1.

120 Chapter 7. The finite element approximation
Convergence of the Lagrange Pk finite element method
We can consider the internal approximation problem of finding uh ∈ V k
0,h such that:
�
Ω
u ′ h (x)v′ �
�
h (x) dx + c(x)uh(x)vh(x) dx =
Ω
f(x)vh(x) dx ,
Ω
∀ vh ∈ V k
0,h . (7.11)
Following the same analysis than for the Lagrange P2 element, we have the following convergence result.
Theorem 7.3 (Convergence) Suppose u ∈ H1 0 (Ω) and uh ∈ V k
0,h
are the solutions of (7.2) and (7.11),
respectively. Then, the Lagrange Pk finite element method converges, i.e. we have:
lim
h→0 �u − uh�H 1 (Ω) = 0 .
Furthermore, if u ∈ H k+1 (Ω) (i.e. f ∈ H k−1 (Ω)), then there exists a constant C independent of h such
that:
�u − uh� H 1 (Ω) ≤ C h k �f� H k−1 (Ω) .
Remark 7.7 1. The approximate solution uh converges toward the exact solution u in H 1 (Ω) when
h → 0. The Lagrange Pk method is of order k in h, if the function f is sufficiently smooth.
2. The matrix assembly becomes more and more difficult as the value of k increases. Since the size of
the problem increases as well, this may result in additional difficulties in solving the resulting linear
system.
3. The computation of the components of the right-hand side vector Fh must be carried out with a
sufficiently accurate method.
7.3 Triangular finite elements in higher dimensions
We consider here a boundary-value problem posed in an open bounded domain Ω ⊂ Rd (d = 2, 3, in
general). For the sake of simplicity, we restrict our study to domains with piecewise polygonal (resp.
polyhedral when d = 3) boundaries, i.e. ¯ Ω can be exactly covered by a finite union of polygons (resp.
polyhedra).
To set the ideas, we consider the homogeneous boundary-value problem (7.2) posed here in an open
bounded domain Ω of R2 :
Given f ∈ L2 (Ω) and c ∈ L∞ (Ω), find u such that:
�
−∆u + cu = f , in Ω
(7.12)
u = 0 , on ∂Ω
We already know that this problem has a unique solution u ∈ H 1 0 (Ω).
7.3.1 Preliminary definitions
We proceed like in one dimension of space. The domain Ω is decomposed into a set of N finite elements,
triangles (Kj)1≤j≤N in dimension d = 2 and tetrahedra in dimension d = 3. These two types of elements
belong to the generic class of simplices.

7.3. Triangular finite elements in higher dimensions 121
d-Simplices
Definition 7.2 A d-simplex K is the convex hull (envelope) of d + 1 points (aj)1≤j≤d+1 in R d , called
the vertices of K, that are not all lying in the same hyperplane. It is the smallest convex passing through
all these points.
Remark 7.8 Let consider d + 1 points (aj)1≤j≤d+1 in R d and let denote (ai,j)1≤i≤d the coordinates of
vector (aj). These points are affinely independent, i.e. not lying in the same hyperplane, if the matrix
⎛
a1,1
⎜
⎜a2,1
⎜
M = ⎜ .
⎝ad,1
a1,2
a2,2
.
ad,2
. . .
. . .
. ..
. . .
⎞
a1,d+1
a2,d+1
⎟
. ⎟
ad,d+1⎠
1 1 . . . 1
is invertible. In such case, the simplex is not degenerated. Any d-simplex has the same number of faces
and vertices, each face being itself a d − 1-simplex.
Furthermore, a few geometric parameters characterize a simplex K:
(i) the diameter hK: the length of the largest element edge,
(ii) the roundness ρK: the diameter of the largest inscribed ball,
(iii) the aspect ratio σK = hK
Barycentric coordinates
ρK
: a measure of the non-degeneracy of K.
Any simplex K can be represented by the barycentric coordinates {λj}1≤j≤d+1 of its vertices. Barycentric
coordinates are a form of homogeneous coordinates.
Definition 7.3 For every 1 ≤ j ≤ d+1, the barycentric coordinate λj of a point x ∈ R d is the first-degree
polynomial:
such that for all 1 ≤ i ≤ d + 1, λj(ai) = δij.
λj(x) = c1x1 + · · · + cdxd + cd+1 ,
For each j, the d + 1 coefficients of the barycentric coordinate λj are the unknowns of a linear system of
d + 1 equations. All d + 1 systems share the same matrix M t and give a unique solution if the simplex
K is not degenerated.
Proposition 7.2 For every point x ∈ R d , there exists a unique vector (λj(x))1≤j≤d+1 such that the
following identites hold:
�d+1
x = ajλj(x) , and
j=1
�d+1
λj(x) = 1 .
j=1

122 Chapter 7. The finite element approximation
Figure 7.5: Example (left) and counter-example (right) of conforming triangulation, in two dimensions.
Proof. For each point x = (xi)1≤i≤d, the scalar values (λj(x))1≤j≤d+1 are the solutions of a d + 1 × d + 1 linear
system that admits M as asociated matrix. Hence, there is a unique solution to this system if the simplex K is non
degenerated. Each function λj is an affine function and one can check easily that λj(ai) = δij will be a solution of
this system. Since there is only one such affine function, it is then the barycentric coordinates function. �
Since the λj are affine functions of x, then we can write:
K = {x ∈ R d , 0 ≤ λj(x) ≤ 1 , 1 ≤ j ≤ d + 1} ,
the faces of K are the intersections of K with the hyperplans λj(x) = 0, 1 ≤ j ≤ d + 1. We observe
also that the change from Cartesian coordinates to barycentric coordinates is an affine transformation.
Hence, a polynomial of total degree k in Cartesian coordinates can be expressed as a polynomial of total
degree k in barycentric coordinates, and conversely. For a first-degree polynomial p we have:
Triangulations and meshes
�d+1
p(x) = p(aj)λj(x) . (7.13)
j=1
Definition 7.4 A triangulation, also called a triangular mesh of ¯ Ω is a set Th of non degenerated dsimplices
(Kj)1≤j≤N such that:
(i) Kj ⊂ ¯ Ω and ¯ Ω =
N�
Kj,
j=1
(ii) the intersection Ki ∩ Kj of any two simplices is a m-simplex, 0 ≤ m ≤ d − 1, such that all its
vertices are also vertices of Ki and Kj.
This definition states that the intersection of two triangles, if it is not empty, shall be reduced to either
a common vertex or an edge. Similarly, in three dimensions, the intersection of two tetrahedra can be
either empty, or reduced to a single common entity (vertex, edge or face). Such a mesh is often called a
conforming mesh (cf. Figure 7.5). The vertices or nodes of the mesh Th are the vertices of the d-simplices
Kj that compose the mesh. Algorithms to construct such triangulations will be described in Chapter 9
and we refer the reader to [Frey-George, 2000] for more information on this topic.
We introduce two conditions on the geometry of a triangulation, with respect to the diameter and
the roundness of its elements.

7.3. Triangular finite elements in higher dimensions 123
Definition 7.5 Suppose (Th)h>0 is a sequence of meshes of Ω. This sequence is said to be a sequence of
regular meshes, or a quasi-uniform sequence, if:
1. the sequence h = max hK tends toward 0,
K∈Th
2. there exists a constant C ≥ 1 such that:
∀h > 0 , ∀K ∈ Th ,
h
ρK
≤ C . (7.14)
Remark 7.9 In dimension two, if K is a triangle, the condition 7.14 is equivalent to the existence of
an angle θ0 > 0 such that
∀h > 0 , ∀K ∈ Th , θK ≥ θ0 ,
where θK is the smallest vertex angle in triangle K.
A set of points of a simplex K has a specific role as defined hereafter.
Definition 7.6 For every k ∈ N∗ , we call principal lattice of order k the set:
�
�
Σk = x ∈ K , λj(x) ∈ 0, 1
�
�
2 k − 1
, , . . . , , 1 , for 1 ≤ j ≤ d + 1 . (7.15)
k k k
For k = 1, the lattice is simply the set of vertices of K; for k = 2, the principal lattice is composed of
the vertices and of the midpoints of the edges (Figure 7.6). More generally, a lattice Σk is a finite set of
points (σj)1≤j≤Nk .
Polynomial spaces
We introduce the set Pk of the polynomials p with scalar coefficients of Rd in R of degree less than or
equal to k:
⎧
⎪⎨
Pk = p(x) =
⎪⎩
�
αi1,...,id
i1,...,id≥0
i1+...id≤k
xi1 1 . . . xid
d , αij ∈ R , x = (x1,
⎫
⎪⎬
. . . xd) .
⎪⎭
Hence, in two and three dimensions of space, we will simply denote:
⎧
⎨
�
Pk = p(x, y) = αijx
⎩
0

124 Chapter 7. The finite element approximation
It is easy to verify that Pk is a vector space of dimension:
⎧
k + 1 d = 1
k�
� � � � ⎪⎨
d + l − 1 d + k
1
dim(Pk) =
= =
(k + 1)(k + 2) d = 2
l
k
2
l=0
⎪⎩
1
(k + 1)(k + 2)(k + 3) d = 3
6
The notion of lattice Σk of a simplex K allows to define a bijective mapping between a space of polynomials
Pk and a set of points (σj)1≤j≤Nk . The set Σk is said to be unisolvent for Pk. We will use this property
to define a finite element.
Lemma 7.8 Given a simplex K. For k ≥ 1, we consider the lattice Σk of order k whose points are
denoted (σj)1≤j≤Nk . Then, every polynomial p ∈ Pk is uniquely determined by its values at the points
(σj)1≤j≤Nk . There exists a basis (ϕj)1≤j≤Nk of Pk such that:
ϕj(σi) = δij , 1 ≤ i, j ≤ Nk .
Proof. At first, we notice that the cardinal of Σk and the dimension of the vector space Pk coincide:
card(Σk) = dim(Pk) =
Indeed, we can write the elements of Σk as follows:
⎧
⎨ d� αj
Σk =
⎩ k ai + � d�
1 −
j=1
j=1
αj
k
(d + k)!
d!k!
.
⎫
� ⎬
α0 , 0 ≤ α1 + · · · + αd ≤ k
⎭ ,
where the αj ∈ N. We know that the mapping associating to every polynomial Pk its values on the lattice Σk is a
linear mapping. Hence, it is sufficient to show that it is an injection to have the bijective property. We will prove
by recurrence on the dimension d that if p ∈ Pk is such that p(x) = 0 for all x ∈ Σk the p = 0 on R d . At first,
notice that a polynomial of degree k that vanishes in k + 1 points of R is identically null. Suppose this is also true
for the dimension d − 1. We use a recurrence on the degree k. For k = 1, an affine function that vanishes at the
vertices of a non-degenerated simplex K is identically null according to the relation (7.13). Suppose this property
is true for all polynoials of degree k − 1 and let consider a degree k polynomial p that vanishes on Σk. We observe
that Σk contains the subset
Σ ′ k = {x ∈ Σk , λ0(x) = 0} ,
that corresponds to the principal lattice of order k of the d−1-simplex of vertices (a1, . . . , ad). Since the restriction
of p to the hyperplane generated by (a1, . . . , ad) is a polynomial of degree k in d − 1 variables, then p = 0 on this
hyperplan, tnaks to the recurrence hypothesis. If we introduce a system of coordinates (x1, . . . , xd) such that the
hyperplane is now defined by xd = 0, then
p(x!, . . . , xd) = xdq(x1, . . . , xd−1) ,
where q is a polynomial of degree d − 1 that vanishes on the set Σk − Σ ′ k since xd is non null on this set. The set
Σk − Σ ′ k is a principal lattice of order k − 1 and thus the recurrence hypothesis leads to conclude that q = 0 and
consquently that p = 0. The results follows. �
In practice, we consider only polynomials of degree 1 or 2. Equation (7.13) provides the characterization
of a polynomial of degree one. Given a d-simplex K of vertices (aj)1≤j≤d+1, we define the edge
midpoints (ajj ′)1≤j

7.3. Triangular finite elements in higher dimensions 125
The principal lattice Σ2 is exactly composed of the vertices and the edge midpoints and every polynomial
p ∈ P2 can be written as:
�d+1
p(x) = p(aj)λj(x)(2λj(x) − 1) +
j=1
�
1≤j

126 Chapter 7. The finite element approximation
Proof. It is easy to see that the elements of V k h belong to H1 (Ω). The Lemma 7.8 allozs to conclude that each
function vh ∈ V k h is exactly known by assembling on each Ki ∈ Th polynomials of degree k that coincide on the
degrees of dreedom of the d-faces. By assembling the basis of Pk on each Ki, the basis (ϕj)1≤j≤Ndof is defined. �
Corollary 7.4 The subspace V k
0,h is a subspace of H1 0 (Ω) of finite dimension corresponding to the number
of internal degrees of freedom, i.e.] not taking the nodes on ∂Ω into account.
Definition 7.8 The triad (K, Pk, Σk) is said to be unisolvent if and only if the mapping v ∈ Pk ↦→
(ϕ1(v), . . . , ϕNdof (v)) ∈ RNdof is an isomorphism.
The unisolvency property is equivalent to say that every function in the polynomial space Pk is entirely
determined by its node values.
7.3.3 Finite element approximation of a boundary-value problem
We return to the numerical approximation of the solution of the homogeneous Dirichlet boundary-value
problem (7.12). The variational formulation of the internal approximation reads:
find uh ∈ V k
0,h such that
�
�
�
(∇uh · ∇vh)(x) +
Ω
c(x)uh(x)vh(x) dx =
Ω
f(x)vh(x) dx ,
Ω
∀vh ∈ V k
0,h , (7.17)
By decomposing uh on the canonical basis (ϕj)1≤j≤Ndof and considering as test functions vh = ϕi, we
obtain:
Ndof �
j=1
��
�
� �
uh(aj) (∇ϕj · ∇ϕi)(x) dx + c(x)ϕj(x)ϕi(x) dx = f(x)ϕi(x) dx .
Ω
Ω
Ω
This formulation leads to solving a linbear system in R Ndof :
AhUh = Fh ,
where we have introduced the notations Uh = (uh(aj))1≤j≤Ndof and Fh = ( �
��
�
Ah = (∇ϕj · ∇ϕi)(x) dx +
Ω
Ω
�
c(x)ϕj(x)ϕi(x) dx
Ω fϕi dx)1≤i≤Ndof and
1≤i,j≤Ndof
It is easy to see that the matrix Ah can be decomposed as a sum of a stiffness matrix Kh and a mass
matrix Mh. Actually, this result is independent of the dimension.
Coefficients of Ah
Since the shape functions ϕj have a small support around a node ai, the intersection of the supports of
ϕj and ϕi if often the emptyset and thus the resulting matrix Ah will contains lot of zero coefficients. It
is a sparse matrix. The coefficients of Ah can be computed via an exact integration formula. Let denote
(λi(x))1≤i≤d+1 the barycentric coordinates of the point x in a simplex K ∈ Th. For every (αi)1≤i≤d+1 we
have: �
λ1(x)
K
α1 . . . λd+1(x) αd+1
α1 + . . . αd+1! d!
dx = |K| � � � ,
αj + d !
1≤j≤d+1
.

7.3. Triangular finite elements in higher dimensions 127
a 1
a 13
a3
Figure 7.7: Nodes of the degrees of freedom on a finite element (K, P2, Σ2).
P = P1 P2 Σ = { p(ai), 1 ≤ i ≤ 3 } Σ = { p(ai), 1 ≤
where |K| i ≤= 3; mes(K) p(aij), denotes 1 ≤ i < jthe ≤ 3 ”volume” } of the simplex K. The integrals of the right-hand side term
P1
Fh are approximated using quadrature formulas in each simplex K ∈ Th. For instance the generalization
K λi(m) 1 ≤ i ≤ 3
of the trapeze formula for a simplex K is:
λi(m) = 1 − aim
ai
m = (x, y)
· ni
�
, j �= i,
aiaj · ni
ϕj(x) dx ≈
K ni
λi(m)
K
j
i j1 j2 i aj1aj2 · ni = 0
|K| �d+1
ϕj(ai) ,
d + 1
i=1
where (ai) ≤i+. 1 represent the vertices of K. These formulas are exact for affine functions. For instance,
λi(m) ai aj
considering a triangle K ∈ R
ak
(ajak)
2 of vertices (ai)1≤i≤3 and denoting by (aij)1≤i,j≤3 the midpoints of the
edges aiaj (cf. Figure 7.7), the following quadrature formula is exact for ϕk ∈ P2:
P2
a 12
a 23
�
ϕk(x) ≈
λi(2λi − 1), 1 ≤ i ≤ 3, 4λiλj, 1 ≤ i < j ≤ 3.
K
|K| �
ϕk(aij) ,
3
1≤i,j≤3
Ω ⊂ R2 k = 1
Th N V k h (ϕ1, . . . , ϕN )
A ∈ RN,N �
Aij = ∇ϕi · ∇ϕj = �
where |K| represents the area of simplex K, while the following formula is exact for ϕk ∈ P3:
�
ϕk(x) dx ≈
K
�
∇ϕi · ∇ϕj, 1 ≤ i, j ≤ N.
|K|
⎛
⎝
60
�
ϕk(ai) + 8
1≤i≤3
�
⎞
ϕk(aij) + 27ϕk(a0) ⎠ ,
1≤i

128 Chapter 7. The finite element approximation
â3
â1
�K
â2
FK
Figure 7.8: Affine transformation of the reference linear triangle in two dimensions.
where BK is a d × d matrix such that the column i is given by the coordinates of ai − a0. Since the
simplex K is non-degenerated, BK is invertible and FK is a one-to-one mapping that maps � K on K and
we have:
|K| = |det(BK)| | � K| = |det(BK)|
.
d!
We use the notation ˆ· in the reference element. Moreover, to simplify, we denote ˆq any quantity
obtained by transporting a quantity q using the transformation FK. Hence, we denote:
ˆx = F −1
K (x) = B−1
K (x − a0) ⇔ x = FK(ˆx) .
For every function v defined on K, we define ˆv defined on � K as:
a1
K
a3
ˆv(ˆx) = v(x) ⇔ ˆv = v ◦ FK = v(FK(ˆx)) ,
and we remove the ˆ· notation when dealing with F −1
K , for every function ˆv defined on � K, we write:
v(x) = ˆv ◦ F −1
K
−1
(x) = ˆv(F (x)) .
Similarly, if ψ is a linear form acting on the functions defined on K, we define the transported linear
form ˆ ψ acting on the functions defined on � K as:
ˆψ(ˆv) = ψ(v) .
Notice that the barycentric coordinates are preserved by the affine trasnformation FK:
ˆλi(ˆx) = λi(x) .
This leads to conclude that the principal lattice of order k of the simplex K is the image by FK of the
principal lattice of order k of � K, denoted by � Σk. Finally, we observe that the space of polynoials of degree
k is left invariant by the transformation FK.
Proposition 7.3 We denote by � · � the Euclidean norm of R d and its subordinate norm. Hence, we
have:
�BK� ≤ hK
, �B
ρ bK
−1
K � ≤ hKb , |det(BK)| =
ρK
|K|
| � K| .
K
a2

7.3. Triangular finite elements in higher dimensions 129
Proof. by definition,
�BK� = sup
v∈Rd �BKv� �BKv�
= sup ,
�v� �v�=ρ ρ cK Kb
and the sup is attained. Let v ∈ R d be a vector of norm ρ b K that correspond to the sup. Hence, there exists two
points ˆy and ˆz on the boundary of the inscribed sphere of � K such that v = ˆy − ˆz. Thus,
where y, z are tzo points in K. Hence,
BKv = BK(ˆy − ˆz) = FK(ˆy) − FK(ˆz) = y − z
�BKv� = �y − z� ≤ hK .
The second inequality is deduced from the first by interchanging the roles of K and � K. The third equality is
obtained by noticing that |det(BK)| is the Jacobian of FK. Hence,
� �
|K| = dx = |det(BK)| dˆx = |det(BK)| | � K| .
K
bK
�
As a consequence, we deduce that there exists two constants C1 and C2, independent of K such that:
Transformation of the derivatives
C2ρ d K ≤ |det(BK)| ≤ C1h d K .
We have the following identities:
�
�v�Lp (K) = |v(x)|
K
p �
dx = |v(FK(ˆx))|
bK
p �
|det(BK)| dˆx
= d! |K| |ˆv(ˆx)| p ,
dˆx
thus yielding the identity:
bK
�v� L p (K) = C |K| 1/p �ˆv� L p ( b K) , (7.18)
with C = (d!) 1/p . Moreover, we observe that if v(x) = ˆv(ˆx), i.e. if ˆv = v ◦ FK, then we have:
∇ˆv(ˆx) = B t K∇v(x) ,
and we obtain the identity:
|v| 2 H1 (K) =
�
�∇v(x)�
K
2 �
dx = �(B
bK
t K)‘−1∇ˆv(ˆx)� 2 |det(BK)| dˆx
= C2 d! |K|
ρ2 �
�∇ˆv(ˆx)�
K
bK
2 ,
dˆx
and then:
|v| H 1 (K) ≤ C hK
|K| 1/2 ρK|ˆv| H 1 ( b K) , (7.19)
where C depends only on the dimension d. By interchanging the role of K and � K, we obtain similarly:
We give the following generalization result.
|ˆv| H1 ( K) b ≤ C |K|1/2
|v| H1 (K) . (7.20)
ρK

130 Chapter 7. The finite element approximation
Theorem 7.4 If v ∈ H m (K), then ˆv ∈ H m ( � K) and there exists a constant C1, independent of K, such
that:
∀v ∈ H m (K) , |ˆv| H m ( b K) ≤ C1�BK� m |det(BK)| −1/2 |v| H m (K) . (7.21)
If ˆv ∈ H m ( � K), then v ∈ H m (K) and there exists a constant C2, independent of K, such that:
∀ˆv ∈ H m ( � K) , |v| H m (K) ≤ C2�B −1
K �m |det(BK)| 1/2 |ˆv| H m ( b K) . (7.22)
Proof. Result admitted here. �
Remark 7.12 Now, we are able to construct finite elements from a unisolvent triad ( � K, � Pk, � Σk) by
defining directly the finite element (K, Pk, Σk) as the transport of these quantities by the transformation
FK. It is easy to see that the new finite element is unisolvent.
7.3.5 Convergence of the finite element method
We introduce a preliminary result.
Proposition 7.4 Let consider (Th)h>0 a sequence of regular triangulations of ¯ Ω ⊂ R d (d ≤ 3). Then,
for every v ∈ H k+1 (Ω), the interpolate rhv is defined and there exists a constant C, independent of h
such that:
�v − rhv� H 1 (Ω) ≤ C h k �v� H k+1 (Ω) .
The following result states the convergence of the Pk finite element method.
Theorem 7.5 Let consider (Th)h>0 a sequence of regular triangulations of ¯ Ω ⊂ R d (d ≤ 3) where all
elements in all triangulations are affine equivalent to a same reference element ( � K, � Pk, � Σk) of class C 0
for a given k ≥ 1 such that Pk ⊂ � P ⊂ H 1 ( � K). Then, the Pk finite element method converges, i.e.
the approximate solution uh of the problem (7.17) converges toward the solution u of problem (7.12) in
H 1 (Ω):
lim
h→0 �u − uh� H 1 (Ω) = 0 .
Furthermore, if u ∈ H k+1 (Ω), then there exists a constant C independent of h such that:
�u − uh� H 1 (Ω) ≤ C h k �u� H k+1 (Ω) ,
Proof. To show the convergence, we use Theorem (4.11) where we consider V = C ∞ c (Ω) ⊂ H k+1 (Ω) dense in
H 1 (Ω). The estimate in the previous proposition allows to verify the assumption of Theorem (4.11). Then, we use
Céa’s lemma to write:
�u − uh�H 1 (Ω) ≤ C inf
vh∈V k
�u − vh�H 1 (Ω) ≤ C�u − rhu�H 1 (Ω) ,
0,h
if rhu ∈ H 1 (Ω). The previous proposition allows to conclude. �
Remark 7.13 This result involves the exact solution uh of the internal approximation problem in V k
0,h .
This requires computing all integrals in Ah and Fh exactly. Since numerical integration formulas are used
in practice, this result may not be valid. Nonetheless, if these integration formulas are exact or at least
very accurate, the order of convergence of the Lagrange finite element method is preserved,

7.3. Triangular finite elements in higher dimensions 131
The Theorem 7.5 shows that the regularity of the exact solution has a direct impact on the order
of convergence of the finite element approximation. Hence, it is often important to have an a priori
knowledge of its regularity. The following result gives an indication for a convex domain.
Theorem 7.6 Let Ω be a convex polygon and let f ∈ R2 . Then, the solution u of the homogeneous
Dirichlet problem:
find u such that: �
�
∇u.∇v =
Ω
fv ,
Ω
∀v ∈ H 1 0 (Ω) ,
is in H2 (Ω) and we have the following estimate:
∀f ∈ L 2 (Ω) , �u� H 2 (Ω) ≤ �f� L 2 (Ω) .
7.3.6 Numerical resolution of the linear system
We remind here that the finite element approximation of the boundary-value problem (7.12) implies the
numerical solving of a linear system in R Ndof :
where Uh = (uh(aj))1≤j≤Ndof and Fh = ( �
is of the form:
Ah Uh = (Kh + Mh) Uh = Fh ,
Ω fϕi dx)1≤j≤Ndof
kij = a(ϕj, ϕi) , 1 ≤ i, j ≤ Ndof .
. The generic term of the stiffness matrix Kh
where a(·, ·) is the bilinear form defined on the functional space H1 0 and (ϕj)1≤j≤Ndof is the canonical
basis of the approximation space. We have mentioned already that the stiffness matrix Kh inherits of
the properties of the bilinear form a. More precisely, if a is V-elliptic, then Kh is positive definite and
if a is symmetric then Kh is symmetric. We have also indicated that the matrix Ah is a sparse matrix,
i.e. it contains a large number of zeroes, as well as the matrices Kh and Mh. Hence, iterative methods
are preferred to solve this linear system (cf. Chapter 5).
Let denote 0 < λ1(Kh) < · · · < λNdof (Kh) the eigenvalues of Kh and 0 < λ1(Mh) < · · · < λNdof (Mh)
the eigenvalues of Mh. We introduce the notations:
and we have the following result.
λNdof (Kh)
λNdof (Mh)
κ(Kh) = , and κ(Mh) = ,
λ1(Kh)
λ1(Mh)
Proposition 7.5 Suppose (Th)h>0 is a quasi-uniform sequence. Then, there exists two constants C1 > 0,
C2 > 0 and a constant C3 > 0 such that:
C1 ≤ h 2 κ(Kh) ≤ C2 , and κ(Mh) ≤ C3 .
We observe that the mass matrix Mh is always well-conditioned since its condition number κ(Mh) is
independent of the mesh size h, under the condition that the refinement is quasi-uniform. On the other
hand, the stiffness matrix Kh becomes ill-conditioned, i.e. κ(Kh) ≫ 1, when the mesh size h is small.
Remark 7.14 We have shown that the approximation error between the exact solution u and the Galerkin
finite element solution uh is bounded by above by a constant times the ”distance” between he spaces V
and Vh. The smaller the distance, the better the approximation error will be. Is seems then natural to
increase the dimension of the approximation space Vh in order to improve the accuracy of the solution.
To this end, we can either:

132 Chapter 7. The finite element approximation
1. increase the number of elements and thus increase the global number of degrees of freedom while
preserving the local reference element. This strategy is known as h-refinement,
2. increase the degree of the polynomials and thus modifying the local reference element while preserving
the total number of elements. Obviously, this choice can be only be applied if the exact solution u
is suffciently smooth. This strategy is known as p-refinement.
We will discuss these adaptation options and others in the chapter 9.
7.4 The finite element method for the Stokes problem
To illustrate the application of the finite element method to solve a system of partial differential equations,
we consider the Stokes problem in two dimensions of space.
We consider the flow of a fluid inside a bounded domain Ω ⊂ R2 and subjected to an external force
field f. The flow is considered as stationary and inertial forces are supposed negligible here, hence the
Stokes equations introduced in Chapter 4 describe this viscous flow problem:
Find (u, p) in appropriate Hilbert spaces such that:
⎧
⎪⎨ −ν∆u + ∇p = f in Ω
div u = g in Ω
(7.23)
⎪⎩
u = 0 on ∂Ω
where the unknowns u and p represent the velocity and the pressure of the fluid, respectively and ν > 0
is the viscosity. Usually, in the applications, the flow is incompressible and the function g is equal to
zero.
7.4.1 Mixed formulation
We have already seen the mixed formulation of this problem in Chapter 4. We introduce the functional
space:
L 2 0(Ω) = {q ∈ L 2 �
(Ω) , q dx = 0} ,
and the mixed formulation reads:
Given f ∈ L 2 (Ω) and g ∈ L 2 0 (Ω), find u ∈ H1 0 (Ω)2 and p ∈ L 2 0
�
where we posed a(u, v) = ν
Ω
(Ω) such that:
� a(u, v) + b(v, p) = f(v) , ∀ v ∈ H 1 0 (Ω) 2 ,
b(u, q) = g(q) , ∀ q ∈ L 2 0(Ω)
�
�
∇u : ∇v, b(v, p) = − p div v, f(v) =
Ω
Ω
Ω
functional spaces H1 0 (Ω)2 and L2 (Ω) 2 are endowed with the canonical norms:
⎛
�u�H1 (Ω) = ⎝ �
i=1,2
�ui� 2 H 1 (Ω)
for u = (ui)1≤i≤2 ∈ H 1 0 (Ω)2 and f = (fi)1≤i≤2 ∈ L 2 (Ω) 2 .
⎞
⎠
1/2
⎛
, �f�L2 (Ω) = ⎝ �
i=1,2
�
fv and g(q) = −
�fi� 2 L 2 (Ω)
⎞
⎠
1/2
,
Ω
(7.24)
gq. The
Remark 7.15 If f ∈ C 0 ( ¯ Ω) 2 and g ∈ C 0 ( ¯ Ω), if Ω is of class C 2 and if the solution (u, p) of the
problem (7.24) is such that u ∈ C 2 ( ¯ Ω) 2 and p ∈ C 1 ( ¯ Ω), then (u, p) is the classical solution of the Stokes
problem.

7.4. The finite element method for the Stokes problem 133
The formulation above is the not the only weak formulation possible for the Stokes problem. An alternate
view consists in including the divergence constraint (for instance the incompressibility constraint div u =
0) directly in the space of test functions. To this end, we introduce the space:
V0 = {v ∈ H 1 0 (Ω) 2 , div v = 0} .
Since the divergence operator is surjective, there exists a function ug ∈ H1 0 (Ω)2 g ∈ L
such that for every
2 0 (Ω), we have:
div ug = g , and �ug�H 1 (Ω) 2 ≤ c�g�L2 (Ω) ,
where c > 0 is a constant (see [Brezis, 2005] for more details). Introducing the change of variables
u ′ = u − ug leads to the condition div u ′ = 0 and to consider the following weak formulation:
find u ′ ∈ V0 such that:
a(u ′ , v) = 〈f, v〉 − a(ug, v) , ∀v ∈ V0 . (7.25)
This weak formulation is called a constrained formulation, since the restriction on the test functions to
be in the space V0 leads to discard the term b(v, p) that vanishes here.
Proposition 7.6 The problem (7.25) is a well-posed problem.
Proof. It is a direct application of Lax-Milgam theorem. � Although this new formulation has some
theoretical advantages, many practical drawbacks prevent its application for solving a discrete problem.
It is indeed difficult to construct divergence-free finite elements. The reader interested is referred to the
book [Brezzi-Fortin, 1991] for more details on this topic.
7.4.2 Numerical approximation
We will now investigate the condition required for the approximation spaces in velocity and pressure to
define a well-posed discrete problem.
Let consider Xh a subspace of the space H1 0 (Ω)2 and Mh a subspace of the space L2 0 (Ω). We introduce
the discrete Stokes problem:
Find uh ∈ Xh and ph ∈ Mh such that:
⎧ �
�
�
⎪⎨ ν ∇uh : ∇vh − ph div vh = f · vh , ∀vh ∈ Xh ,
Ω
�Ω
�Ω
(7.26)
⎪⎩
We introduce the space
Ker(Bh) = {vh ∈ Xh ,
qh div uh =
Ω
and we enounce the fundamental compatibility result.
�
gqh , ∀qh ∈ Mh .
Ω
qh div vh = 0 , ∀qh ∈ Mh} ,
Ω
Theorem 7.7 The discrete problem (7.26) is a well-posed problem if and only if the spaces Xh and Mh
satisfy the following compatibility condition:
∃βh > 0 , inf
qh div vh
Ω
sup
qh∈Mh vh∈Xh �qh�L2 (Ω)�vh�H 1 (Ω)
�
≥ βh . (7.27)

134 Chapter 7. The finite element approximation
Furthermore, under this condition, we have the following estimates:
�u − uh�H 1 (Ω) ≤ c1h inf �u − vh�H 1 (Ω) + c2h inf �p − qh�L2 (Ω)
vh∈Xh
qh∈Mh
�p − ph�L2 (Ω) ≤ c3h inf �u − vh�H 1 (Ω) + c4h inf �p − qh�L2 (Ω)
vh∈Xh
qh∈Mh
where the constants (cih)1≤i≤4 are given by Theorem 4.12.
(7.28)
(7.29)
Proof. We know that the bilinear form a(·, ·) is V -elliptic on H 1 0 (Ω) 2 × H 1 0 (Ω) 2 and the result is then a direct
consequence of Theorem 4.12. �
The following result provides a practical method for checking whether the compatibility condition is
satisfied or not.
Lemma 7.10 (Fortin) Let consider the spaces X = H1 0 (Ω)2 and M = L2 0 (Ω) and let b ∈ L(X, M).
Suppose there exists β > 0 such that:
Then, there exists βh > 0 such that:
inf
inf
q∈M sup
v∈X
sup
qh∈Mh vh∈Xh
b(v, q)
�v�X�q�M
b(vh, qh)
�vh�Xh �qh�Mh
≥ β .
≥ βh ,
if and only if there exists γh > 0 such that for every v ∈ X, there exists a restriction operator rh ∈
L(X, Xh), i.e. rh(v) ∈ Xh, such that:
∀qh ∈ Mh , b(v, qh) = b(rh(v), qh) and �rh(v)�X ≤ γh�v�X .
Proof. If the criterion is satisfied, for every qh ∈ Mh we write
b(vh, qh)
sup
vh∈Xh �vh�X
b(rh(v), qh)
≥ sup
v∈X �rh(v)�X
≥ 1 b(v, qh)
sup
γh v∈X �v�X
≥ β
�qh�M .
γh
Conversely, if the discrete condition is satisfied, then the operator Bh : Xh → Mh defined by b(vh, qh) = 〈Bhvh, qh〉
defines an isomorphism (admitted here). Hence, for every v ∈ X, there exists a unique vh = rh(v) such that
The mapping v ↦→ rh(v) is linear and we have:
∀qh ∈ Mh , b(rh(v), qh) = b(v, qh) .
�rh(v)�X ≤ 1
�Bhrh(v)�M ′
β
≤ 1
β sup
b(rh(v), qh)
qh∈Mh �qh�M
≤ 1
β sup
b(v, qh)
ah∈Mh �qh�M
≤ γh
β �v�X .

7.4. The finite element method for the Stokes problem 135
The operator rh has the desired properties. �
This proof provides a technique to verify the discrete compatibility condition; it consists in constructing
the operator rh using the solution of the system:
b(rh(v), qh) = b(v, qh) , ∀qh ∈ Mh ,
and to check if it is bounded in the space X. This is clearly is non-trivial task. On the other hand, it
is possible to construct a simpler operator for specific approximation spaces and finite elements and to
check if the compatibility condition is satisfied.
It is easy to construct an internal variational approximation of the Stokes problem, as we have already
seen:
find uh ∈ Xh and ph ∈ Mh such that:
�
a(uh, vh) + b(vh, ph) = f(vh) , ∀vh ∈ Xh ,
(7.30)
b(uh, qh) = g(qh) , ∀qh ∈ Mh ,
If we denote by nu (resp. np) the dimension of the space Xh (resp. Mh), we introduce the basis (ϕj)1≤j≤nu
of Xh and the basis (ψj)1≤j≤np of Mh defined using the basis shape functions of the finite elements and
we decompose uh and ph on these basis:
nu
np
�
�
uh = uh(aj)ϕj(x) , and ph(x) = ph(a ′ j)ψj(x) .
j=1
We obtain the following linear system to solve:
�
Ah Bt � � �
h
Uh
=
Bh 0
Ph
� Fh
Gh
j=1
�
. (7.31)
where Uh = (uh(aj))1≤j≤nu and Ph = (ph(a ′ j ))1≤j≤np and the matrices Ah = (aij) ∈ R nu,nu and Bh =
(bki) ∈ Rnp,nu are given by:
�
�
aij = ν ∇ϕi : ∇ϕj dx , and bki = −
Ω
The vectors Fh = (fj)1≤j≤nu and Gh = (gk)1≤k≤np
�
fi =
Ω
are defined as:
�
f · ϕi dx , and gk =
Ω
ψi div ϕk dx .
Ω
g ψk dx .
Lemma 7.11 The vector 1 ∈ Rnp is always contained in the kernel Ker(Bt h ). The discrete pressure ph
is defined up to an additive constant.
Proof. Let rh ∈ Mh and wh ∈ Xh. By definition,
Since rh = 1 is in Mh and since
Wh · B t hRh = BhWh · Rh =
∀wh ∈ Xh ,
�
Ω
�
rh div wh dx .
Ω
�
div wh dx = wh · n ds = 0 .
∂Ω
we conclude that Rh = 1 ∈ Ker(Bt h ). �
Finding a pair of approximation spaces (Xh, Mh) for the Stokes problem has been an active research
field in numerical analysis. We present here a few examples of such pairs.

136 Chapter 7. The finite element approximation
The P1/P1 finite element
This seems the obvious first choice for implementing the finite element method for the Stokes problem.
We consider the unit domain Ω =]0, 1[ 2 and we suppose a regular Cartesian mesh is defined on Ω (Figure
7.9). We introduce the approximation spaces:
Xh = {uh ∈ C 0 ( ¯ Ω) 2 , ∀K ∈ Th , uh|K ∈ P 2 1 , uh|∂Ω = 0} ,
Mh = {ph ∈ L 2 0(Ω) ∩ C 0 ( ¯ Ω) , ∀K ∈ Th , ph|K ∈ P1} .
For a given element K ∈ Th, we denote (aiK)1≤i≤3 its vertices. Then, we have:
∀vh ∈ Xh ,
�
ph div vh =
Ω
�
�
(div vh)|K ph ,
K∈Th
K
= �
3� |K|
(div vh)|K
3
K∈Th
j=1
ph(aj,K) ,
Now, let us consider the pressure field ph defined on the triangulation Th by its values 0, −1, 1 at the
three vertices of every triangle K ∈ Th. In this case, we have
and it is easy to see that:
3�
ph(aj,K) = 0 ,
j=1
∀vh ∈ Xh ,
�
ph div vh = 0 .
Ω
This is clearly in contradiction with the requirement of the inf-sup condition. Hence, the P1/P1 finite
element shall be discarded for solving the Stokes problem.
When the dimension of Ker(B t h ) is strictly larger than one, the finite element method is unstable. In
such case, solving the linear system using an iterative method will result in oscillations in the pressure
values. On the other hand, if dim(Ker(B t h )) = 1, the discrete pressure can be uniquely determined by
fixing its value at a mesh node or by setting its average value on the domain Ω.
1
0
-1
1
-1
1
0
-1
0 1
1
-1
1
-1 0
Figure 7.9: A regular Cartesian mesh of the unit domain Ω =]0, 1[ 2 and the unstable pressure mode for
the P1/P1 finite element.
0
0
-1
1
0
-1
1
0
-1
1

7.4. The finite element method for the Stokes problem 137
The P1/P0 finite element
This finite element presents the a priori advantage of being very simple to implement. Given a triangulation
of the domain Ω, we define a piecewise continuous approximation of the velocity and a piecewise
constant (and thus discontinuous) approximation for the pressure. Since the velocity is piecewise affine,
its divergence is piecewise constant and thus by testing the divergence using constant functions, the
divergence of the discrete field is strongly forced to be null. Hence, this finite element allows to obtain
exactly a divergence free velocity field. Nonetheless, the P1/P0 finite element does not fulfill the inf-sup
condition and Fortin’s criterion.
Again, we provide a counter-example in two dimensions. Suppose the tiangulation Th of the simply
connected polygonal domain Ω contains t triangles, vi internal vertices and vb boundary vertices. We
shall have 2vi degrees of freedom for the space Xh, as the velocity must vanish on the boundary and t
degrees of freedom for the pressure leading to t − 1 independent divergence-free constraints. The Euler
formulas for a simple polygon indicate that:
from which we deduce that:
t = 2vi + vb − 2 ,
(t − 1) ≥ 2(vi − 1) .
We observed that dim(Mh) = t − 1 and that dim(Xh) = 2vi. The rank theorem allows to write:
dim(Ker(B t h )) = dim(Mh) − dim(Im(B t h )) ≥ (t − 1) − dim(Xh)
= (t − 1) − 2vi = vb − 3
This last relation indicates that there is at least vb − 3 spurious pressure modes. In other words, the
space Mh is too rich to impose the Bhuh = 0 constraint. A function uh ∈ Xh is thus overconstrained and
a locking phenomenon occurs. In general, uh = 0 is the only discrete divergence-free function of Xh such
that Bhuh = 0; i.e. Bh is not surjective.
The P1-bubble/P1 (mini) finite element
We have seen that with the P1/P1 finite element, the dimension of the space of the degrees of freedom
of the velocity is not large enough. To overcome this problem, it seems natural to attempt enriching the
velocity space rather than impoverishing the pressure space. However, it is not necessary to consider a
space of polynomials of degree two for approximating the velocity. Crouzeix and Raviart suggested (in
1973) to add one extra degree of freedom to the barycenter of every simplex of the triangulation Th of
the domain Ω. Hence, we define in two dimensions:
P1 = (P1(K) ⊕ Span{bK , K ∈ Th}) 2 ,
with the bubble function bK defined in two dimensions as:
bK = 27
3�
i=1
�
λi ,
λi , and generalized in any dimension as bK = (d + 1) d+1
d+1
where (λi)1≤i≤3 denote the barycentric coordinates on K. The degrees of freedom associated with P1 are
defined as:
�
Σk = {v ↦→ v(ai) , 1 ≤ i ≤ 3} ∪ {v ↦→
Then, we consider the approximation spaces:
ei
v · ni dσ} .
i=1

244 Chapitre 11
138 Chapter 7. The finite element approximation
Vitesse Pression
Figure 11.1 – Élément Mini (O(h))
Figure 7.10: The P1-bubble/P1 (mini) element for velocity (left) and pressure (right) approximation.
Mh = {ph ∈ L 2 0(Ω) ∩ C 0 ( ¯ Ω) , ∀K ∈ Th , ph|K ∈ P1} .
Vitesse
Pression
Xh = {vh ∈ C 0 ( ¯ Ω) 2 , ∀K ∈ Th , vh|K ∈ P1 , uh|∂Ω = 0} ,
(7.32)
It is easy to see that Xh ⊂ (H 1 0 (Ω))2 .
Before giving a compatibility result for this finite element, we need to introduce an interpolation
operator rh. Since v ∈ H 1 0 (Ω)2 ) is not necessarily continuous, its Lagrange interpolate might not be
defined. Therefore, in each triangle, we define rh as:
rh(v) = Ch(v) + αK bK ,
Figure 11.2 – Élément Mini en dimension 3 (O(h))
liberté en vitesse et 3 en pression dans le cas bidimensionnel (respectivement 15 et 4 en dimension
3). Cet élément est très utilisé, particulièrement en dimension 3 (voir la figure 11.2) en raison de
son faible nombre de degrés de liberté. Cet élément converge à l’ordre 1 car l’approximation en
vitesse ne contient que les polynômes de degré 1.
Le deuxième élément illustré est celui de Taylor-Hood souvent appelé P2 − P1 en raison de
l’approximation quadratique des composantes de vitesse et de l’approximation P1 de la pression
(voir les figures 11.3 et 11.4. Les noeuds de calcul sont situés aux sommets (vitesses et pression) et
aux milieux des arêtes (vitesses seulement). Cet élément converge à l’ordre 2 (O(h2 where Ch(v) = (Ch(v1), Ch(v2)) is the Clément interpolate defined hereafter and αK ∈ R
)) en vertu du
théorème 11.14 car l’approximation en vitesse contient les polynômes de degré 2 et l’approximation
en pression contient ceux d’ordre 1. Des équivalents quadragulaires et hexaédriques, également
d’ordre 2, sont illustrés aux figures 11.5 et 11.6.
Remarque 11.4
Dans le cas d’éléments à pression continue, il est parfois nécessaire d’imposer une condition supplémentaire
sur le maillage pour avoir l’existence et l’unicité de la solution. On devra par exemple
supposer qu’aucun élément n’a deux côtés sur la frontière du domaine (voir les exercices de fin de
chapitre). �
2 is chosen so as
to satisfy the equation: �
(rh(v) − v) = 0 .
K
To this end, we define:
�� �−1 �
αK = bK (v − Ch(v)) .
K
K
For every v ∈ Xh and qh ∈ Mh, we write:
�
�
qh div(v − rh(v)) = ∇qh(rh(v) − v) =
Ω
Ω
�
�
∇qh(rh(v) − v) = 0 ,
K
K∈Th
since ∇qh is piecewise constant on every K ∈ Th.
Definition 7.9 Let Th be a triangulation of Ω, Γh be the set of the internal vertices of Th. We introduce
Θh the space of continuous piecewise linear functions on Th which are zero on the boundary and we
denote (ϕγ)γ∈Γh the basis of P1 finite elements with homogeneous boundary conditions. We introduce the
operator Ch ∈ L(H1 0 (Ω), Θh), proposed by Clément in 1975, and defined as follows:
Ch(v) = �
� � �
1
v ϕγ ,
|ωγ|
where the set ωγ is defined as the union of all triangles K ∈ Th for which γ is a vertex.
γ∈Γh
In this definition, the nodal value v(γ) is replaced in the definition of the interpolant rh by the local
average value around γ. The Clément operator can be applied to any function in L 2 (Ω). We have the
following result.
Lemma 7.12 If the family of triangulations (Th)h>0 is quasi-uniform, then the operator Ch satisfies the
local estimates:
∀v ∈ H 1 0 (Ω) , �v − Ch(v)� L 2 (K) ≤ C hK |v| H 1 (ωK) ,
ωγ

7.4. The finite element method for the Stokes problem 139
∀v ∈ H 1 0 (Ω) , |Ch(v)| H 1 (K) ≤ C |v| H 1 (ωK) ,
where ωK is the union of all triangles sharing at least one of the vertices of K.
Theorem 7.8 If the family of triangulations (Th)h>0 is quasi-uniform, the approximation spaces Xh and
Mh are uniformely compatible with respect to h, i.e. there exists a constant βh, independent f h, satisfying
the condition (7.27).
Proof. Let consider v ∈ H 1 0 (Ω) 2 . We construct the operator rh(v) ∈ Xh such that
∀qh ∈ Mh ,
�
K∈Th
qh div(rh(v)) =
Ω
Since Mh ⊂ H1 (Ω), the previous identity is equivalent to writing:
∀qh ∈ Mh ,
�
�
rh(v) · ∇qh = �
�
Since ∇qh ∈ Pd 0, the previous identity leads to conclude that:
� �
∀K ∈ Th , rh(v) =
K
K
�
qh div(v) .
Ω
K∈Th
K
v .
K
v · ∇qh .
Since the operator rh is composed of two terms, we analyze them separately. Regarding the Clément interpolate
Ch, we invoke the previous estimate:
For the term αKbK, we write:
∀v ∈ H 1 0 (Ω) , |Ch(v)| H 1 (K) ≤ C |v| H 1 (ωK) .
|αKbK| H 1 (K) ≤ C|αK| ,
with the constant C independent of hK and K. Then, we write
��
|αK| = bK
K
≤ C 1
|K| |
�
� −1
K
�
| (v − Ch(v))|
K
(v − Ch(v))|
≤ |K| −1/2 �v − Ch(v)� L 2 (K) .
We use the estimate �v − Ch(v)� L 2 (K) ≤ C hK |v| H 1 (ωK) , to obtain:
and finally:
|αK| ≤ C |v| H 1 (ωK) ,
|rh(v)| H 1 (K) ≤ C |v| H 1 (ωK) ,
By taking the square of this expression and summing over all K ∈ Th we conclude that:
�rh(v)� H 1 (Ω) ≤ C�v� H 1 (Ω) .
The final result is obtained by invoking Fortin’s lemma. �
Proposition 7.7 Suppose the solution (u, p) of the problem (7.24) is sifficiently smoth, i.e. u ∈ H 2 (Ω) 2 ∩
H 1 0 (Ω)2 and p ∈ H 1 (Ω) ∩ L 2 0 (Ω). Then, the solution (uh, ph) of the problem (7.26) with the spaces defined
by (7.32) is such that:
�u − uh� H 1 (Ω) + �p − ph� L 2 (Ω) ≤ C h (�u� H 2 (Ω) + �p� H 1 (Ω)) . (7.33)

140 Chapter 7. The finite element approximation
7.4.3 Numerical resolution
We have seen that the discrete problem (7.30) leads to solve the linear system (7.31). We assume here
the natural hypothesis that the bilinear form a(·, cdot) is V -elliptic on X with a constant α and that the
form b(·, ·) satisfies on xh × Mh a inf-sup condition uniformely in h, and we denote the constant β.
The resolution of the linear system using a direct method is likely to be costly because of the number
of degrees of freedom and thus the size of the linear system. Direct methods are appropriate up to
Ndof = 104 . For this reason, iterative methods are favored. However, the matrix of the system (7.31)
is neither positive nor definite. It is indeed an indefinite matrix (but symmetric) with that zero block
on the diagonal. It will have positive and also negative eigenvalues. It is possible to make it positive by
solving the system: �
AhU + B t hP = Fh
. (7.34)
−BhU = −Gh
However, the new matrix corresponding to this system is still indefinite and is no longer symmetric. We
know from Chapter 5 that gradient methods are not efficient on this type of matrix. We present two
commonly used methods for solving such mixed problems.
Resolution of the saddle point by penalization
The following method is gaining popularity for solving the Stokes problem (and saddle point problems
in general) because of its versatility and facility to be implemented. It consists in replacing the problem
(7.31) by the following problem:
�
AhUε + B t hPε = Fh
, (7.35)
−BhUε + εShPε = −Gh
where the matrix Sh ∈ R np,np is such that:
(ShP, Q) = 〈ph, qh〉M ,
where (·, ·) and 〈·, ·〉M denote the standard Euclidean scalar product in R np and the inner product in M,
respectively. In the system (7.35), ε is a penalization coefficient sufficiently small.
The main idea is simple. Suppose Ah is a positive definite symmetric matrix, the system (7.31) is
then equivalent to:
�
1
inf
BhVh=Gh 2 (AhVh,
�
Vh) − (Fh, Vh) .
If Sh is any positive definite matrix in Rnp,np , the previous expression can be replaced by:
�
1
inf
Vh 2 (AhVh, Vh) + 1
2ε (S−1
h (BhVh
�
− Gh, BhVh − Gh) − (Fh, Vh) ,
or equivalently, by writing Pε = (1/ε)S −1
h (BhVh − Gh):
� AhUε + B t h Pε = Fh
BhUε − Gh = εShPε
. (7.36)
By replacing the quantity Pε in the first equation, we obtain the linear system:
�
Ah + 1
ε Bt hS−1 h Bh
�
Uε = Fh + 1
ε Bt hS−1 h Gh . (7.37)

7.4. The finite element method for the Stokes problem 141
If the matrix S −1
h is a sparse matrix, this provides an efficient technique to solve our problem. This system
can be solved by an iterative technique like the conjugate gradient, since the matrix is now positive definite
and symmetric. Notice however that the term 1/εBt hS−1 h Bh has a strong negative impact on the condition
number of the linear system (7.4.3).
We now investigate the effect of changing the constrained problem to a penalized problem. We have
the following error estimate.
Proposition 7.8 Let consider ε > 0 and let (U, P ) and (Uε, Pε) denote the solutions of system (7.34)
and (7.36), respectively. Then, we have the error estimate:
αβ�U − Uε�X + αβ 2 �P − Pε�M ≤ Cε�P �M . (7.38)
Proof. By difference, we obtain the system:
� Ah(U − Uε) + B t h(P − Pε) = 0
−Bh(U − Uε) − εShPε = 0 .
We introduce on R nu the norm � · �∗ defined by:
∀U ∈ R nu , �U�∗ = sup
V ∈Rnu (U, V )
.
�V �X
The continuity and the V -ellipticity of the bilinear form a implies that:
∀U ∈ R nu , �AhU�∗ ≤ �a� �U�X , and ∀U ∈ R nu , α�U�X ≤ �AhU�∗ .
The matrix Bh ∈ R np,np satisfies the inf-sup condition:
or equivalently:
The continuity of the form b implies that:
min
�P �M �=0 max
(B
�U�X �=0
t hP, U)
�P �M �U�X
≥ β ,
∀P ∈ R np , β�P �M ≤ �B t hP �∗ .
∀P ∈ R np , �B t hP �∗ ≤ �b� �P �M .
Using these inequalities leads to write the first equation of the system as:
�P − Pε�M ≤ 1
β �Bt h(P − Pε)�∗
= 1
β �Ah(U − Uε)�∗ ≤ C
β �U − Uε�X .
By multiplying the first equation by (U − Uε) and using the V -ellipticity of the form a we obtain for the second
equation:
α�U − Uε� 2 X ≤ (Ah(U − Uε), U − Uε) = (B t h(Pε − P ), U − Uε) = (Pε − P, Bh(U − Uε)) = −ε(Pε − P, ShPε)
= −ε(Pε − P, Sh(Pε − P )) − ε(Pε − P, ShP )
≤ −ε(Pε − P, ShP )
≤ ε�Pε − P �M �P �M .
The error estimate is obtained by combining the two previous inequalities. �
Remark 7.16 Notice that discretizing a penalized problem is not necessarily equivalent to penalize a
discrete problem [Brezzi-Fortin, 1991]. The penalty method is considered in this last case as a procedural
(algorithmic) technique, since a choice of spaces Xh ⊂ X and Mh ⊂ M has been done. On the other
hand, discetizing the penalized problem consists in choosing Qh = BhVh that is in general a poor choice.

142 Chapter 7. The finite element approximation
Uzawa’s operator
To conclude this section, we introduce an iterative, although very efficient, algorithm to solve such
problem when th operator Ah associated with the bilinear form a is invertible. We know since Chapter
4 that the Stokes equations are equivalent to solving a problem of energy minimization. More precisely,
the resolution of the linear system (7.31) is equivalent to the following minimization:
J(Uh) = min J(Vh) with J(Vh) =
Vh∈KerBh
1
2 (AhVh, Vh) − (Fh, Vh) . (7.39)
In the optimization terminology, the pressure is the Lagrange multiplier that imposes the constraint
div u = g when the energy is minimized.
First, we eliminate the variable Uh from the linear system (7.31), if the matrix Ah is invertible, thus
leading to:
Uh = A −1
h (Fh − B t hP )
and then by replacing Uh in the second equation BhUh = Gh, we obtain the system:
We consider the Uzawa matrix U = BhA −1
h Bh.
BhA −1
h Bt h Ph = BhA −1
h Fh − Gh . (7.40)
Proposition 7.9 If matrix Ah is positive definite, this matrix is also positive definite.
Proof. One has:
It is positive definite if KerB t h
(BhA −1
h Bt hP, P ) = (A −1
h Bt hP, B t hP ) ≥ α�B t hP � .
= {0}. �
Hence, problem (7.40) is more easy to solve than the original problem (7.31) as efficient methods exist for
positive definite systems. Unfortunately, even if the matrix Ah is a sparse matrix (i.e. contains lot of zero
values), its inverse A −1
h is likely to be a full matrix. The resolution of (7.40) must be performed using an
iterative method since inverting the matrix A −1
h is extremely computationally costly. The convergence
rate of iterative methods is strongly related to the condition number of the matrix.
Proposition 7.10 If the matrix U is symmetric, we have the estimate:
κ(U) ≤ C
α 2 β 2 κ(Sh) .
Remark 7.17 If the family of mesh (Th)h>0 is quasi-uniform, the condition number κ(Sh) is a constant
independent of h. Hence, the previous estimate shows that the conditioning of Uzawa’s operator is of order
one (if α and β are of order one, which is a reasonable hypothesis). The iterative techniques converge
fast without any preconditioning.