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Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation

In the previous lecture we have introduced the following space
of functions
defined on (0, 1):
(1)
V =
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0

(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization
problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and
contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡

∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1,
M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity
of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following
theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following
inequalities are
valid:

(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO
LAZAROV
Proof: We give two proofs. The simple one proves the above
inequali-

ties with C0 = 1/2 and C1 = 1. The better proof establishes the
above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and
apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣ ∣ ∣ ∫ x
0
u′(s)ds
∣ ∣ ∣ 2 ≤ ∫ x
0

1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this
inequality
we get the first inequality (3) with C1 = 1. Further, increasing
the r.h.s by
taking the integral in the whole interval and then integrating the
obtained
inequality for x ∈ (0, 1) we get the second inequality (3) with
C0 = 1/2.
This simple proof uses only one boundary condition u(0) = 0.
We can
improve the constants if we use also the second boundary
condition u(1) = 0.
Namely, we derive in the same manner the inequality

(5) |u(x)|2 = |
∫ 1
x
u′(s)ds |2 ≤
∫ 1
x
1ds
∫ 1
x
(u′(s))2 ≤ (1 −x)
∫ 1
x
(u′(s))2ds.
Now one multiplies (4) by 1 −x and (5) by x and adds the two
inequalities
to get the estimate:
|u(x)|2 ≤ x(1 −x)
∫ 1

0
(u′(s))2ds.
This will allows us to show (3) with C0 = 1/6 and C1 = 1/4,
correspondingly.
The appropriate inequalities for an arbitrary l are obtained by
change of
the variable.
Now the coercivity of the bilinear form follows easily from
these inequal-
ities and the assumptions (2). Indeed,
a(u,u) ≥ k0
∫ 1
0
u′2dx ≥
k0
2
∫ 1

0
(u′2 + u2)dx ≥
k0
2
‖u‖2V .
2. Abstract form of Ritz-Galerkin method
Instead of (V ), we shall consider its approximation. Namely, let
Vh be a
n-dimensional subspace of V . We consider the following
simpler problem:
(Vh) find uh ∈ Vh such that a(uh,v) = L(v), ∀ v ∈ Vh.
One can show that this problem is equivalent to the following
minimization
problem in Vh:
(Mh) find uh ∈ Vh such that F(uh) ≤ F(v), ∀ v ∈ Vh.
There some simple but important for the applications properties
that
are easily obtaind from the equivalence of the problems (Vh)
and (Mh).

Using the fact that a(u,u) = L(u) and a(uh,uh) = L(uh) we get
from the
inequality F(u) ≤ F(uh) that
1
2
a(u,u) −L(u) ≤
1
2
a(uh,uh) −L(uh) =⇒ −
1
2
a(u,u) ≤−
1
2
a(uh,uh)
LECTURE # 3: ABSTRACT RITZ-GALERKIN METHOD 3
which gives a(u,u) ≥ a(uh,uh). This inequality has a clear

physical inter-
pretation.
Since Vh is n-dimensional, we can assume that it is spanned by
n linearly
independent functions φj(x) ∈ V, j = 1, ...,n; i.e.
Vh =
n∑
j=1
cjφj(x), cj are arbitrary constants
We may relate the parameter h to n by h = 1/n.
Examples of Spaces Vh:
Example 1:

φj(x) = sin(jπx), j = 1, ...,n. This will produce the so-called
spectral
method.
Example 2:
φj(x) = xj(1 −x), j = 1, ...,p. The so-called p-version of FEM.
Example 3:
The construction of the space is done in the following manner:
split the
interval [0, 1] into n + 1 subintervals by introducing the points
xj = jh,
j = 0, ...,n + 1, where h = 1
n+1
. The space Vh consists of all continuous
functions on [0, 1] that are linear on each subinterval (element)
[xj−1,xj]
and vanish at x = 0 and x = 1. Obviously, the functions in the
space Vh are
determined by their values at the nodes xj, j = 1, ...,n. Solving

(Vh) with
such space Vh will lead to the finite element method.
The following set of functions can serve as a basis for Vh:
(6) φj(x) =
x−xj−1
h
, x ∈ [xj−1,xj];
xj+1 −x
h
, x ∈ [xj,xj+1];
0, elsewhere ;
j = 1, ...,n. The functions are constructed in such way that φj(x)
is 1 at
the node xj, 0 at all remaining nodes, and linear over the finite
elements.

This basis is called a nodal basis. Note, that this is just one
possible basis.
Another example is the so-called hierarchical basis.
Obviously, the solution uh of the problem (Vh) is in the form
uh(x) =
n∑
j=1
ξjφj(x), where ξj are unknown constants.
LAZAROV
0 1x
j
j
φ

Figure 1. A nodal basis function for linear finite elements
Then the method (Vh) can be written in the form
a(uh,v) = l(v), ∀ v ∈ Vh =⇒ a
j=1
ξjφj(x),φk
(7)
This produces a linear system called also
(Ritz or Galerkin system) for the unknown ξ ∈ Rn:
n∑
j=1
ξja(φj,φk) = L(φk), k = 1, ...,n, in matrix form Aξ = b,

where A ≡{ajk}nj,k=1 = {a(φj,φk)}
n
j,k=1, is a square n×n matrix
and b = {L(φj)}nj=1, and ξ = {ξj}
n
j=1 are vector-columns in R
n.
The matrix A is often called “stiffness” matrix while b is the
“load” vector
which is computed from the data. Since the bilinear form a(., .)
is coercive
the matrix A is nonsingular (show this) and therefore the system
Aξ = b
has unique solution for any b. However, the condition number
of A play an
importnat role in the numerical methods for solving the system
and there
is a necessity to discuss this in details.
3. Mixed boundary conditions
For the boundary value problem

(D)
−(k(x)u′)′ + q(x)u = f(x), in (0, 1)
u(0) = 0,
u(1) + k(1)u′(1) = β1
we introduced the space V :
(8) V =
v(x) is continuous function on (0, 1);
v′(x) exists in a generalized sense and is in L2(0, 1);
v(0) = 0
and the variational formulation (V ) with
a(u,v) ≡
∫ 1

0
(k(x)u′v′ + q(x)uv) dx + u(1)v(1)
and
L(v) ≡
∫ 1
0
f(x)v dx + β1v(1).
Examples of Spaces Vh for the problem (D):
Example 1:
φj(x) = sin((j − 0.5)πx), j = 1, ...,n. This will produce the so-
called
spectral method.
Example 2:

φj(x) = xj, j = 1, ...,p. This will produce the so-called p-version
of
Galerkin method.
Example 3: (the finite element method)
The construction of the space is done in the following manner:
split the
interval [0, 1] into n subintervals by introducing the points xj =
jh, j =
0, ...,n, where h = 1
n
. The space Vh consists of all continuous functions on
[0, 1] that are linear on each subinterval (element) [xj−1,xj] and
vanish at
x = 0. Obviously, the space Vh is determined by the values of a
function at
the nodes xj, j = 1, ...,n. Solving (Vh) with such space Vh will
lead to the
finite element method.
In this case, a nodal basis will consist of all functions from the

previous
example corresponding to internal nodes as well as one more
function that
will involve the value at end xn = 1: The following set of
functions can serve
as a basis for Vh:
(9) φn(x) =
x−xn−1
h
, x ∈ [xn−1,xn];
0, elsewhere.
Another possible basis in Vh is the so-called hierarchical basis
which utilizes
hierarchy of grids.
4. Neumann boundary conditions

Now we shall consider the following simple model problem for
the un-
known function u(x):
(D)
−u′′ + u = f(x), in (0, 1)
u′(0) = 0.
u′(1) = 0.
In the previous lecture we have introduced the set of functions
defined on
(0, 1) that are is continuous function, have piece-wise
continuous deriva-
tive.This set has been equipped with the norms
||v||2 = (v,v) and ||v||2V = (v,v)V = (v,v) + (v
′,v′).
LAZAROV
After completing the set V in the norm || · ||V we get the

Sobolev space
H1(0, 1) of functions having generalized first derivatives in
L2(0, 1). Note,
that the functions in V do not satisfy any boundary conditions.
Therefore,
V ≡ H1(0, 1).
We also introduced the following variational problem:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
where
a(u,v) ≡
∫ 1
0
(u′v′ + uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
We shall study the Ritz system for this particular BVP. It is
obvious, that
a(u,v) = (u,v)V so this form is trivially coercive.

We have introduced the following finite dimensional space:
split the in-
terval [0, 1] into n− 1 subintervals by introducing the points xj
= (j − 1)h,
j = 1, ...,n, where h = 1
n−1 ; the space Vh consists of all continuous on [0, 1]
functions that are linear on each subinterval (element) [xj−1,xj].
Obviously,
the functions in the space Vh can be determined by their values
at the nodes
xj, j = 1, ...,n. The approximate problem (Vh) for such space Vh
will lead
to the finite element method with linear elements.
The following set of functions can serve as a basis for Vh:
(10) φj(x) =
x−xj−1
h

, x ∈ [xj−1,xj];
xj+1 −x
h
, x ∈ [xj,xj+1];
0, elsewhere ;
j = 2, ...,n− 1 and two additional functions defined at the end-
points:
(11)
φ1(x) =
x2 −x
h
, x ∈ [x1,x2];
0, elsewhere ;
φn(x) =

x−xn−1
h
, x ∈ [xn−1,xn];
0, elsewhere .
The functions are constructed in such way that φj(x) is 1 at the
node xj,
0 at all remaining nodes and linear over the finite elements.
This basis is
called nodal basis.
The solution uh of the problem (Vh) is in the form
uh(x) =
n∑
j=1
ξjφj(x), where ξj are unknown constants.

Then the method (Vh) can be written in the form (7).
This basis of the space Vh will produce a tridiagonal matrix A.
Indeed,
a(φj,φk) = 0, for |j −k| > 1. Also, for j = k we get
a(φj,φj) =
∫ xj+1
xj−1
(
1
h2
+ φ2j (x)
)
dx =
2
h
+
2h
3
, for 1 < j < n,

a(φ1,φ1) =
∫ x2
x1
(
1
h2
+ φ21(x)
)
dx =
1
h
+
h
3
, for j = 1,
a(φn,φn) =
∫ xn
xn−1

(
1
h2
+ φ2n(x)
)
dx =
1
h
+
h
3
, for j = n.
Similarly, for k = j + 1 we get
a(φj,φj+1) =

∫ xj+1
xj
(
−1
h2
+ φj(x)φj+1(x)
)
dx =
−1
h
+
h
6
.
The coefficients below the main diagonal are recover from the
symmetry of
the matrix A.
Thus, the matrix A of the Ritz-system has the form A = A0 +

A1, where:
(12) A1 =
1
h
1 −1 0 . . . 0
−1 2 −1 . . . 0
0 −1 2 . . . 0
. . . . . . .
0 0 0 . . . 1
= h6
2 1 0 . . . 0
1 4 1 . . . 0
0 1 4 . . . 0

. . . . . . .
0 0 0 . . . 2
Matrix A1 is called “stiffness” matrix, while the matrix A0 is
called “mass”
matrix. Both matrices are symmetric and A0 is positive definite
while A1 is
semi-definite.
5. Issues to be addressed
In the genral case we are facing the following issues:
• to assemble the matrix A and to solve the system Aξ = b;
• alternatively, if an iterative method is used that requires only
the
matrix-vector multiplication Aξ, then one should prepare a pro-
cedure of matrix vector multiplication (possibly without
explicitly
forming the matrix A);
• estimate the condition number of the matrix A for a prticular
chice

of the basis of the space Vh;
• estimate the error e = u−uh;
• to develope an algorithm that adaptively choses the mesh so
that
the error is uniformly distributed in the domain and is dreven
below
a desired level.
We need to develop the mathematical tools for studying these
problems.
This includes: estimate for the condition number of A,
deriving/finding fast
methods for solving the system, proving various integral
inequalities, deriv-
ing the approximation error with piece-wise polynomial
functions, estimates
in various Sobolev norms, etc.
6. An estimate of the condition number of the global matrix A
for Neumann BC
Further, we shall use the following definition of a condition
number of a

symmetric and positive definite matrix:
cond(A) =
max λ(A)
min λ(A)
,
where λ(A) is an eigenvalue of A, i.e. Aξ = λξ, for some ξ a
nonzero vector
in Rn.
Often it is not possible to compute the eigenvalues and the
condition
number, but for practical purposes it is enough to have an upper
bound for
cond(A). For this we need upper and lower bounds for the
eigenvalues of A.
LAZAROV
Simple calculations show that

h
6
≤ λ(A0) ≤ h and 0 ≤ λ(A1) ≤
4
h
.
So we produce the following bound from above for the
condition number of
the matrix A
(13) cond(A) ≤
max λ(A0) + max λ(A1)
min λ(A0) + min λ(A1)
≤
4/h + h
h/6
=
24
h2

+ 6 = O(h−2).
Remark 1. Note, that A0 and A1 are square matrices of size n
and one
finds that cond(A0) ≤ 6 i.e. the condition number of A0 does
not depend
on the size of the matrix. Such matrices are called well-
conditioned. In
contrast, A1 has condition number O(h−2) which increases
quadratically,
when h → 0. Such matrices are called ill-conditioned.
7. Exercises
The following matrices play essential role in the finite element,
finite vol-
ume and finite difference methods for two-point boundary value
problems
and the solution of the corresponding linear systems. The
spectral properties
of these matrices are used very often in the computational
practice.
(1) Find the exact eigenvalues of the matrices B1,B0 ∈ Rn×n
given by

(14) B1 =
2 −1 0 . . . 0 0
−1 2 −1 . . . 0 0
0 −1 2 . . . 0 0
. . . . . . . .
0 0 0 . . . −1 2
and
(15) B0 =
4 1 0 . . . 0 0
1 4 1 . . . 0 0

0 1 4 . . . 0 0
. . . . . . . .
0 0 0 . . . 1 4
Hint: Show that λj(B1) = 4 sin2
πj
2(n+1)
, j = 1, . . . ,n and then use
the fact that B1 +B0 = I, where I is the identity matrix in Rn.
From
these calculations follow that both B1 and B0 are positive
definite.
(2) Estimate that eigenvalies of the scaled “stiffness” matrix B1
∈ Rn×n
(16) B1 =

1 −1 0 . . . 0 0
−1 2 −1 . . . 0 0
0 −1 2 . . . 0 0
. . . . . . . .
0 0 0 . . . −1 1
and the scaled “mass” matrix B0 ∈ Rn×n
(17) B0 =
2 1 0 . . . 0 0
1 4 1 . . . 0 0
0 1 4 . . . 0 0
. . . . . . . .
0 0 0 . . . −1 2

Remark 2. Using the technique applied above we can show that
in this case
the eigenvalues are λj(B1) = 4 sin2
πj
2(n−1) , j = 0, . . . ,n− 1.
Remark 3. The eigenvalues and eigenvectors of these algebraic
problems
and problems obtained by approximation of the same
differential operator
with third type boundary conditions could be found in the
monograph of
Samarskii [6, pp. 104–109].
References
[1] L. C. Evans, Partial Differential Equations, Graduate
Studies in Mathematics, vol.

19, AMS, 1998.
[2] Ch. Grossmann, H.-O. Ross, and M. Stynes, Numerical
Treatment of Partial Differ-
ential Equations, Springer, Berlin, 2005.
[3] M. Renardy and R. Rogers, An Introduction to Partial
Differential Equations, Texts
in Applied Mathematics, Springer-Verlag, 1993.
[4] P. Knabner and L. Angermann, Numerical Methods for
Elliptic and Parabolic PDEs,
Springer-Verlag, New Yrok Inc, 2003.
[5] S. Larsen and V. Thomee, Partial Differential Equations
with Numerical Methods,
Springer, 2003.
[6] A.A. Samarskii, The Theory of Difference Schemes,
Monographs and Textbooks in
Pure and Appled Mathematics, Marcel Dekker, Inc, New York,
2001.
Lecture_08_S08.pdf

LECTURE # 8:
MULTIDIMENSIONAL SECOND ORDER ELLIPTIC
PROBLEMS
MATH610: NUMERICAL METHODS FOR PDES – R.
LAZAROV
1. Introduction and preliminaries
First, we introduce some notations that will be used further.
Here Ω will
denote a polygonal bounded domain in Rd, d = 2, 3 with
boundary ∂Ω.
Further, for the vector q = (q1, . . . , qd) and for a scalar
function v we define
the divergence ∇ · q and the gradient ∇ v, correspondingly, by
∇ · q = ∂q1
∂x1
+ · · · + ∂qd
∂xd

and ∇ v =
(
∂v
∂x1
, . . . ,
∂
∂xd
)
.
The Stokes theorem will be used in the following form:
∫
∂Ω
q · n ds =
∫
Ω

∇ · q dx.
Here, n is the outward unit vector to ∂Ω and q·n denotes the
inner product
of two vectors on Rd.
We shall use the Hilbert space H1(Ω) of functions defined on Ω
and having
their generalized derivatives in L2(Ω). The subspace of those
functions in
H1(Ω) that vanish on the boundary ∂Ω will be denoted by H10
(Ω). The
L2 and H1-inner products of these spaces and the corresponding
norms are
defined as follows:
(u, v) =
∫
Ω
uv dx, (u, v)1 = (u, v) + (∇ u, ∇ v),(1)
‖u‖ = (u, u)1/2, ‖u‖1 = (u, u)1/21 .(2)
For the elements in the space H1(Ω) we shall use the following
Poincare

inequality:
(3)
∫
Ω
u2 dx ≤ M0
∫
Ω
|∇ u|2 dx
where the constant M0 > 0 does not depend on u.
We shall give proof of this inequality for d = 2.Without loss of
generality,
we can assume that Ω is contained in the unit square Π, i.e. Ω ⊂
Π :=
(0, 1) × (0, 1). Then we can extend a function u ∈ H1(Ω) to Π
by zero
1

2 MATH610: NUMERICAL METHODS FOR PDES – R.
LAZAROV
outside Ω. The extended function is denoted by ū. It belongs to
H10 (Π) and
obviously,
∫
Π
ū2 dx =
∫
Ω
u2 dx.
Next, we write the equality
2
∫
Π

ū2 dx =
∫
Π
{(∫ x1
0
∂
∂x1
ū(ξ, x2) dξ
)2
dx(4)
+
(∫ x2
0
∂
∂x2

ū(x1, ξ) dξ
)2 }
dx
and apply Cauchy-Schwarz inequality to each of the line
integrals:
2
∫
Π
ū2 dx ≤
∫
Π
{
x1
∫ 1
0
(

∂
∂x1
ū(ξ, x2) dξ
)2
(5)
+x2
∫ 1
0
(
∂
∂x2
ū(x1, ξ) dξ
)2 }
dx.
Using Fubini theorem,we get finally:
2

∫
Ω
u2 dx = 2
∫
Π
ū2 dx ≤ 1
2
∫
Π
|∇ū|2 dx(6)
=
1
2
∫
Ω
|∇ u|2 dx,

which is the required inequality with M0 = 1/4. If the domain Ω
is contained
in a rectangle (0, l1)×(0, l2) the required inequality follows by
change of the
variables.
Further, we shall need the following two inequalities valid for
functions in
H1(Ω):
(7)
∫
∂Ω
u2 ds ≤ C‖u‖21,
and
(8)
∫
Ω
u2 dx ≤ C

{∫
Ω
|∇ u|2 ds +
∫
∂Ω
u2 ds
}
.
Here the constant C does not depend on u but depend on the
domain Ω.
One can prove these inequalities for rectangular domains simply
by using
the corresponding estimates from the one-dimensional case. The
proofs are
left as an exercise for this part of the class (see, e.g. [3, 7]).
MULTIDIMENSIONAL ELLIPTIC PROBLEMS 3

2. Problem formulation
In this lecture we shall consider the following Dirichlet
boundary-value
problem: find u(x) such that:
(D)
Lu := ∇ ·
(
−K(x)∇ u + b(x)u
)
+ q(x)u = f (x), x ∈ Ω
u(x) = 0, x ∈ ∂Ω.
where the coefficients K(x), b, q and f are given functions on Ω.
We
assume that Ω is a bounded domain with Lipschitz boundary
∂Ω, K(x) is a
symmetric and uniformly in Ω positive definite matrix and the
coefficients
K(x), b(x), q(x) are measurable and bounded function in Ω.

This is the divergent form of the problem. Quite often second
order
problems are given in the following non-divergent form:
(9)
Lu := ∇ · (−K(x)∇u) + b̃(x)∇ u + q(x)u = f (x), x ∈ Ω
u(x) = 0, x ∈ ∂Ω.
If the vector field b̃(x) is differentiable then these two forms are
equivalent.
In case when b ≡ b̃ and ∇ · b = 0, then these two form coinside.
In some applications this equation describes: (1) deflection of
an elastic
membrane under transverse load f (then K = I, b ≡ 0, q ≡ 0); (2)
the
pressure distribution in a porous media (K is the permeability
tensor, b ≡
0, q ≡ 0); (3) concentration distribution of a chemical in a flow
with velocity
b and absorption coefficient q. The quantity
q(x) = −K(x)∇ u + b(x)u
is often called total flux (mass, thermal, etc) with −K(x)∇ u the

diffusive
part and b(x)u convective part of the flux.
For deriving the variational formulation of this problem we
follow the
standard approach used in the 1-dimensional problems. We
multiply the
differential equation (D) by a test function v ∈ H10 (Ω) and
integrate over Ω:
∫
Ω
(
∇ · (−K(x)∇ u + b(x)u) + q(x)u
)
v dx =
∫
Ω
f (x)v dx.

We use the identity
(
∇ · (−K(x)∇ u + b(x)u)} v = ∇ · {(−K(x)∇ u + b(x)u) v
)
(10)
−
(
− K(x)∇ u + b(x)u
)
· ∇ v,
LAZAROV
so that after applying the Stokes theorem we transform the right
hand side
of the above identity to the form:
∫

∂Ω
(
K(x)∇ u − b(x)u
)
· n v ds +
∫
Ω
(
K(x)∇ u − b(x)u
)
· ∇ v dx.
Now we use the fact that v vanishes on ∂Ω to get
∫
Ω
(
(K(x)∇ u − b(x)u) · ∇ v + q(x)uv

)
dx =
∫
Ω
f (x)v dx.
We rewrite this integral identity in the abstract form
a(u, v) = L(v) ∀ v ∈ H10 (Ω),
where
a(u, v) =
∫
Ω
(
K(x)∇ u · ∇ v − ub(x) · ∇ v + q(x)uv
)
dx

and
L(v) =
∫
Ω
f (x)v dx.
Thus, we have shown that the solution of the problem (D)
satisfies the
following variational problem:
(V ) find u ∈ H10 (Ω) such that a(u, v) = L(v), ∀ v ∈ H10 (Ω) .
So we have reformulated the differential problem (D) in terms
of integral
identity involving the bilinear form a(·, ·) and the linear form
L(·). Again,
we can use the general theoretical framework and Lax-Milgram
theorem to
show the existence and the uniqueness of the solution u ∈ H10
(Ω). We shall
prove that under reasonable conditions on the coefficients the
bilinear form
a(·, ·) is coercive and continuous in V = H10 (Ω) so we can

apply the general
theoretical framework for such problems.
Now we give conditions on the coefficients of the differential
equation (D)
that are sufficient for the coercivity and the continuity of the
bilinear form
a(·, ·):
(C)
ξT K(x)ξ ≥ k0ξT ξ, ∀ ξ ∈ Rd, k0 = const > 0,
q(x) + 1
2
∇ · b(x) ≥ 0, ∀ x ∈ Ω.
Theorem 1. Assume that the conditions (C) are satisfied. Then
the bilinear
form a(·, ·) is coercive and continuous in V , i.e. there are
positive constants
α and C such that
(11)
a(u, u) ≥ α‖u‖21, (coercivity)
a(u, v) ≤ C0‖u‖1 ‖v‖1. (continuity)

Proof: First, we note that
(12) −u b · ∇ v = −1
2
∇ · (bu2) + 1
2
u2 ∇ · b.
Then applying the Stokes theorem and the condition (C) and the
fact that
v vanishes on ∂Ω we get the following for for a(u, u)
a(u, u) =
∫
Ω
(K∇ u · ∇ u + (q + 0.5∇ · b)u2) dx ≥ k0||∇ u||2.
Using Poincare inequality (3) we get the desired result

regarding the coer-
civity. The continuity of the bilinear from is a simple
consequence of the
boundness of the coefficients.
Let Vh be a finite dimensional subspace H10 (Ω). The Ritz-
Galerkin method
can be formulated in the already discussed abstract form:
(Vh) find uh ∈ Vh ⊂ H10 (Ω) such that a(uh, v) = L(v), ∀ v ∈
Vh.
Our goal now is to construct the space Vh and to show how the
Ritz-system
derived from (Vh) is computed and solved.
3. Other types of boundary conditions
Instead of Dirichlet boundary conditions one can put various
other types
of boundary conditions on ∂Ω. Below we give two natural
boundary condi-
tions that are widely used in the applications.
Case b ≡ 0; Then we have diffusion-reaction equation and the

following
Robin condition can be prescribed on the whole boundary ∂Ω or
on part of
it:
(13) K(x)∇ u · n + σ(x)u = g(x) ∀ x ∈ ∂Ω.
Here σ(x) ≥ 0 and g(x) are given functions on ∂Ω. If σ(x) ≡ 0
then this is
the classical Neumann boundary condition. The meaning of this
boundary
condition is that we prescribe the diffusive flux on ∂Ω. If σ(x) ≡
g(x) ≡ 0
then no flux is allowed through ∂Ω. This is typical insulated
boundary (in
thermal problem) or no-flow boundary in porous media
applications.
The weak formulation of this boundary-value problem is
obtained in the
same way as in the case of Dirichlet boundary conditions. Then
after inte-
gration by parts and using the boundary conditions we get that u
∈ H1(Ω)
satisfies the following integral identity:

a(u, v) = L(v) ∀ v ∈ H1(Ω),
where
a(u, v) =
∫
Ω
{K(x)∇ u · ∇ v + q(x)uv} dx +
∫
∂Ω
σu v ds
and
L(v) =
∫
Ω
f (x)v dx +
∫
∂Ω

g(x) v ds.
LAZAROV
Note, that the functions in the solution space doe not satisfy any
boundary
conditions. The bilinear form is coercive in H1(Ω) under the
condition that
σ(x) ≥ σ0 = const > 0 ∀ x ∈ ∂Ω. In fact, it is enough that σ(x) ≥
σ0 > 0
on a part of the boundary with a positive measure. Indeed, we
have
a(u, u) ≥ k0
∫
Ω
|∇ u|2 dx + σ0
∫
∂Ω

u2 ds.
Next, we use the embedding inequality (8) to get the missing
‖u‖2-term in
the coercivity.
Similarly,
|L(v)| ≤
∫
Ω
|f v| dx +
∫
∂Ω
|g v| ds(14)
≤‖f‖‖v‖ +
(∫
∂Ω
|g|2 ds

)1/2 (∫
∂Ω
|v|2 ds
)1/2
.(15)
Finally, we use the estimate (7) to get the required continuity of
the linear
form L(v).
Case b 6≡ 0; This diffusion-convection-reaction equation and
the fol-
lowing boundary conditions are quite natural (together with
Dirichlet BC).
First, split the boundary ∂Ω into two parts: ∂Ω = Γin ∪ Γout,
where
Γin = {x ∈ ∂Ω : b(x) · n(x) < 0},
Γout = {x ∈ ∂Ω : b(x) · n(x) ≥ 0}.
Then the following boundary conditions are natural:
(16)

−K(x)∇ u · n = 0, x ∈ Γout,
−K(x)∇ u · n + u b · n = g(x), x ∈ Γin.
The physical meaning of these boundary conditions is the
following: on the
part of the boundary where the flow enter the domain, i.e. b(x) ·
n(x) < 0
we can prescribe either the function or the total flux q.
Then one gets the following form
(Lu, v) =
∫
Ω
{K(x)∇ u · ∇ v − b · ∇ vu + q(x)uv} dx(17)
+
∫
Γout
b · nu v ds +
∫

Γin
g v ds
Now we define the bilinear from a(·, ·) and the linear form L(·)
as
a(u, v) =
∫
Ω
{K(x)∇ u · ∇ v − b · ∇ vu + q(x)uv} dx +
∫
Γout
b · nu v ds
and
L(v) =
∫
Ω
f vdx +

∫
Γin
g v ds.
Ωb
Γ
out
Γ
out
Γ
out
Γ
out
Γ
in

Γ
in
Γ
in
Figure 1. Domain with inflow Γin and outflow Γout boundaries
One can easily prove that if the coefficients of the differential
equation
satisfy one of the conditions
(A) q(x) +
1
2
∇ · b(x) ≥ c0 = const > 0 ∀ x ∈ Ω,
(B) q(x) +
1
2
∇ · b(x) ≥ 0 and the measure of the set Γin is nonzero,
(C) q(x) +
1

2
∇ · b(x) ≥ 0 and the measure of the set Γout is nonzero,
then the corresponding bilinear form will be coercive in H1(Ω)-
norm.
Indeed, using (12) by Stokes’ theorem
a(u, u) =
∫
Ω
(K∇ u · ∇ u + (q + 0.5∇ · b)u2) dx(18)
− 1
2
∫
Γin
b · nu2ds + 1
2
∫

Γout
b · nu2ds.
which is the same as
a(u, u) =
∫
Ω
(K∇ u · ∇ u + (q + 0.5∇ · b)u2) dx + 1
2
∫
∂Ω
|b · n|u2ds.
Obviously one of the conditions (A) – (C) guarantee the
coercivity of the
bilinear form in H1(Ω)-norm. Then, by Lax-Milgram Theorem,
we get the
desired result about existence and uniqueness of the solution of
the equation
(D) with boundary conditions (16).

For this problem the following maximum principle could be
shown
Theorem 2. ([6, Theorem 31., p. 26]) Consider the differential
operator L
of problem (D) and assume that u ∇ C2(Ω̄) and
Lu ≤ 0 (Lu ≥ 0) in Ω.
LAZAROV
(i) If q = 0, then
max
x∇Ω̄
u(x) ≤ max
x∈ ∂Ω
u(x)
(

min
x∇Ω̄
u(x) ≤ min
x∈ ∂Ω
u(x)
)
(ii) If q ≥ 0, then
max
x∇Ω̄
u(x) ≤ max(max
x∈ ∂Ω
u(x), 0)
(
min
x∇Ω̄
u(x) ≤ min( min

x∈ ∂Ω
u(x), 0)
)
This theorem allows us to study the uniqueness and the stability
of the
solution of the problem (D) in maximum-norm. This is quite
useful in many
applications. However, the natural way to numerically address
this problem
is to use the variational form of the problem (D) and derive
approximation
schemes generated by the finite element method.
4. Abstract Galerkin method
We take Vh is a finite dimensional subspace of V (denote the
dimension
of Vh by n) and consider the variational problem on Vh:
(Vh) find uh ∈ Vh such that a(uh, v) = L(v), ∀ v ∈ Vh.
Let {φj (x)}ni=1 be a basis for Vh, so that for uh, v ∈ Vh we
have:

(19) uh(x) =
n∑
i=1
Uiφi(x) and v(x) =
n∑
i=1
Viφi(x).
The parameters in the vector-column U T = (U1, . . . , Un1)T
are called de-
grees of freedom for the finite element method and are obtained
from the
Galerkin system AU = b. Here the entries of the matrix A are aij
= a(φi, φj )
and the vector-load b has components bj = L(φj ). Since we have
assumed
coercivity of the bilinear from a(·, ·), the matrix A is
nonsingular and there-
fore the Galerkin system has unique solution.
Note that the matrix A is non-symmetric (as long as b 6= 0).

Moreover,
for convection dominated problems that is when b is much
larger than K (in
some norm) and this causes a number of serious problems for
any numerical
method.
Our main goal in this class will construction of appropriate (and
practi-
cally feasible) finite dimensional spaces Vh, the spaces of
piece-wise polyno-
mial functions over a partition of Ω into finite elements. The
basic texts we
shall use in this class are the textbooks of Larsen and Thomée
[6] or [5] and
the monographs of Ciarlet [1], and Ern and Guermond [2].
References
[1] P.G. Ciarlet, The Finite Element Method for Elliptic
Problems, Classics of Applied

Mathematics, v. 40, SIAM, 2002.
[2] A. Ern and J.-L. Guermond, Theory and Practice of Finite
Elements, Series of Applied
Mathematical Sciences v. 159, Springer-Verlag, 2004.
Studies in Mathematics, volume
19, American Mathematical Society, 1991.
[4] D. Kinkaid and W. Cheney, Numerical Analysis.
Mathematics of Scientific Comput-
ing, Third Edition, Brooks/Cole, 2002.
[5] P. Knabner and L. Angermann, Numerical Methods for
Elliptic and Parabolic PDEs,
Springer-Verlag, New Yrok Inc, 2003.
[6] S. Larsen and V. Thomée, Partial Differential Equations
Springer-Verlag, Texts in Applied Mathematics 45, 2003.
in Applied Mathematics 13, Springer-Verlag, 1993.

5. Appendix: Abstract variational problem:
We recall for the previous lectures the following general
framework for
elliptic equations.
Let V be a Hilbert space with an inner product (·, ·)V and
corresponding
norm || · ||V . Let the bilinear form a(u, v) defined on V D × V
and the linear
form L(v) defined on V are such that:
(1) a(u, v) is coercive in V , i.e., there is a constant α > 0 such
that
a(v, v) ≥ α||v||2V , ∀ v ∈ V ;
(2) a(u, v) is continuous, i.e., there is a constant C > 0 such that
a(u, v) ≤ C0||u||V ||v||V , ∀ u, v ∈ V ;
(3) L(v) is continuous in V , i.e., there is a constant Λ > 0 such
that
L(v) ≤ Λ||v||V , ∀ v ∈ V .
The following theorem is a particular case of the well-know

Lax-Milgram
theorem for NON-SYMMETRIC bilinear forms (see e.g. [7] for
the proof
of the symmetric and your lecture notes, Lecture # 7, for non-
symmetric
forms a(·, ·)):
Theorem 3. (Lax-Milgram) Let V be the Hilbert space with an
inner product
(u, v)V and let the conditions (1) - (3) holds true. Then the
problem find
u ∈ V s.t.
(20) a(u, v) = L(v), ∀ v ∈ V
has unique solution u ∈ V . Furthermore, the solution satisfies
the stability
estimate:
‖u‖V ≤
Λ
α
.
Lecture_09_S08.pdf

LECTURE # 9:
INTRODUCTION TO FEM FOR SECOND ORDER
ELLIPTIC EQUATIONS
MATH610: NUMERICAL METHODS FOR PDES – R.
LAZAROV
As introduced before, Ω will be a polygonal bounded domain in
Rd, d =
2, 3 with boundary ∂Ω. Further, for the vector q = (q1, . . . ,qd)
and for
a scalar function v we define the divergence ∇ · q and the
gradient ∇ v,
correspondingly, by
∇ · q =
∂q1
∂x1
+ · · · +
∂qd
∂xd

and ∇ v =
(
∂v
∂x1
, . . . ,
∂
∂xd
)
.
No we consider the following model boundary-value problem:
(D)
find u(x) such that:
Lu := −∇ ·∇ u + u := −∆u + u = f(x), x ∈ Ω
∇ u · n :=
∂u

∂n
(x) = 0, x ∈ ∂Ω.
where a(u,v) =
∫
Ω
{∇ u ·∇ v + uv} dx and L(v) =
∫
Ω
f(x)v dx
The variational formulation of this problem is:
(V )
find u ∈ V := H1(Ω) such that a(u,v) = L(v), ∀ v ∈ H1(Ω),
where a(u,v) =
∫
Ω
{∇ u ·∇ v + uv} dx and L(v) =
∫

Ω
f(x)v dx.
So we have reformulated the differential problem (D) in terms
of integral
identity involving the bilinear form a(·, ·) and the linear form
L(·). Again,
we can use the general theoretical framework and Lax-Milgram
theorem to
show the existence and the uniqueness of the solution u ∈
H1(Ω). Obviously,
the bilinear form a(·, ·) is coercive and continuous in V =
H1(Ω) so we can
apply the general theoretical framework for such problems.
Date: September 20, 2010.
1
LAZAROV
Ω

τ
1
τ
2
τ
3
Figure 1. Left: Triangulation of a polygonal domain; Right:
non-conforming triangulation, which is not allowed in our
current considerations
Let Vh be a finite dimensional subspace H1(Ω). The Ritz-
Galerkin method
can be formulated in the already discussed abstract form:
(Vh) find uh ∈ Vh ⊂ H1(Ω) such that a(uh,v) = L(v), ∀ v ∈ Vh.
Our goal now is to construct the space Vh and to show how the
system of
linear equations derived from (Vh) is computed and solved.
We point out that the boundary conditions of the differential

problem
are not imposed on the functions from the space H1(Ω). This
condition is
weakly contained in the variational formulation itself. It is more
natural to
begin with such a problem, since we do not need to impose any
boundary
conditions on the finite dimensional subspace of H1(Ω).
1. FE partition of the domain and FE spaces
We partition the domain Ω into triangular (tetrahedral) finite
elements τ.
The finite elements τ are considered open sets and we denote
their closure
by τ, i.e. τ = τ ∪ ∂τ. This triangulation is denoted by Th. We
shall
consider conforming types of triangulation, i.e. triangulations
that satisfy
the following conditions:
(1)
(a) τ are disjoint, i.e. τi ∩ τj = ∅ , i 6= j;

(b) τi ∩ τj is either:
(i) a vertex of τi & τj;
(ii) an entire edge of τi & τj;
(iii) empty .
An example of a triangulation of the domain is shown on Figure
1. The
right Figure 1 shows a non-conforming triangulation that is
NOT considered
in our current setting.
FEM FOR ELLIPTIC PROBLEMS 3
P
i
Pj
P
k (Pk)=Uk
τ

u (P )=Uh
u (P )=Uh
uh
i jji
Figure 2. Liner triangular finite element
Together with set of all triangles Th we shall use the sets of all
edges Eh
and the sets of all vertices Vh. Further, we define the set Pm of
polynomials
of degree m with real coefficients:
Pm =
0≤i+j≤m
cijx
i
1 x

j
2, cij are real numbers
Now we consider the simpleast case of linear finite elements
and define
the finite-dimensional subspace Vh ⊂ H1(Ω) in the following
way:
Vh =
{
v : v ∈ C0(Ω), v ∈ P1(τ), for τ ∈ Th
}
.
The functions in Vh can be uniquely determined by its values at
the
vertices of the triangulation Th. We shall use the nodal basis in
Vh. If the
number of the vertices in Vh is N, then we define N linearly
independent

functions φj(x), j = 1, . . . ,N by:
φj(x) =
1 if x = Pj, Pj ∈ Vh;
0 if x = Pk, Pk ∈ Vh, Pk 6= Pj;
linear over each τ ∈ Th.
2. Finite Element Computations
Each function in Vh can be presented in the form:
uh(x) =
N∑
j=1
Ujφj(x), where Uj = uh(Pj), Pj ∈ Vh.
Then the finite element method for the problem (V ) reduces to
solving
the Ritz-Galerkin system of linear equations for the unknown
values UT =

(U1,U2, . . . ,UN ):
(2)
N∑
i=1
Uia(φi,φj) = L(φj), j = 1, . . . ,N, or AU = b.
LAZAROV
The entries of the matrix A are a(φi,φj) and the entries of the
load-vector
b are L(φj), i,j = 1, . . . ,N. Since the bilinear form a(·, ·) is
symmetric and
coercive the matrix A is symmetric and positive definite.
The matrix A of the system (2) is computed element-wise.
Namely, the
contributions of a particular finite element τ to the global
“stiffness” and
“mass” matrices are done by element-wise computations.

In each element we introduce local notations: let the triangle τ
has vertices
Pi, Pj, Pk and let the restrictions of the nodal basis functions to
τ be
denoted again by φi, φj, φk. We denote
Uτ =
Uk
Vk
φk(x)

and similarly for the element functions
∇ Φτ (x) := ∇ Φτ =
∇ φi∇ φj
∇ φk
∂φi
∂x1
∂φi
∂x2
∂φj
∂x1

∂φj
∂x2
∂φk
∂x1
∂φk
∂x2
This allows us to write the following presentations:
uh(x)|τ = ΦTτ Uτ, v(x)|τ = Φ
T
τ Vτ, ∇ uh(x)|τ = ∇ Φ
T
τ Uτ, ∇ v(x)|τ = ∇ Φ
T
τ Vτ,

so that ∫
τ
∇ uh(x) ·∇ v(x) dx =
∫
τ
Vτ
T∇ Φτ∇ ΦTτ Uτ dx := V
T
τ A
1
τUτ,∫
τ
uh(x)v(x) dx :=
∫
τ
V Tτ Φτ Φ
T
τ Uτ dx := V

T
τ A
0
τUτ,
and similarly for the r.h.s.∫
τ
f(x)v(x) dx :=
∫
τ
V Tτ Φτf(x) dx := Vτ
Tbτ.
Here A1e and A
0
e are 3 × 3 matrices, called element “stiffness” and “mass”
matrices, correspondingly, and bτ is a vector of dimension 3,
called the
element load vector.
3. Element Stiffness and mass matrices for linear FE

One gets a very simple formula for these matrices, namely:
A1τ =
∫
τ
∇ φi ·∇ φi ∇ φi ·∇ φj ∇ φi ·∇ φk∇ φj ·∇ φi ∇ φj ·∇ φj ∇ φj ·∇ φk
∇ φk ·∇ φi ∇ φk ·∇ φj ∇ φk ·∇ φk
FEM FOR ELLIPTIC PROBLEMS 5
8
9
i−1
y

j−1
y
j
y
j+1
τ1
τ
2
τ
3
τ
4
τ
5
τ
6

0 1
23
4
5 6
h h
h
h
x x
i
x
i+1
Figure 3. Uniform rectangular grid
and
A0τ =

∫
τ
φkφi φkφj φkφk
∫
τ
φiφi
∫
τ
φiφj
∫
τ
φiφk∫
τ

φjφi
∫
τ
φjφj
∫
τ
φjφk∫
τ
φkφi
∫
τ
φkφj
∫
τ
φkφk
Further, we shall show that in fact the element “mass” matrix

A0τ has very
simple form, namely:
A0τ =
|τ|
12
1 1 2
where |τ| denotes the area of the triangle τ. Obviously, the
elements of the
mass matrix are of order h2, where h is the diameter of the
element τ.
Below we give the nodal basis function associated with the node
(xi,yj)
(note we are using the notations (x,y) instead of (x1,x2)).
For the element τ1, which is a right triangle and the vertices are

ordered
in the following way (P0, P1, P2) (see, Figure 3) we can easily
compute the
element “stiffness” matrix A1τ :
A1τ =
1
2
−1 0 1
Assembling the local matrices will give the global matrix of the
Ritz system.
For example, the equation for the internal point point P0 shown
on Figure
LAZAROV

finite element hφ0(x,y) h
∂φ0(x,y)
∂x
h
∂φ0(x,y)
∂y
τ1 h− (x−xi + y −yj) −1 −1
τ2 h− (y −yj) 0 −1
τ3 h + (x−xi) 1 0
τ4 h + (x−xi + y −yj) 1 1
τ5 h + (y −yj) 0 1
τ6 h− (x−xi) −1 0
Table 1. Analytic presentation of the nodal function at the
node 0 and its derivatives
3 will be
4U0 −U1 −U2 −U4 −U5 +
h2
12

(6U0 + U1 + U2 + U3 + U4 + U5 + U6)(3)
=
∫
Ω
f(x)φ0 dx.
Similarly, assembling the equation for the node P4 that is on the
Neumann
boundary we get the equation:
2U4 −U0 −
1
2
U3 −
1
2
U8+
h2
24
(6U4 + 2U0 + U3 + 2U5 + U8)(4)

=
∫
Ω
f(x)φ4 dx.
References
Studies in Mathematics, v. 19,
American Mathematical Society, 1991.
[2] Ch. Grossmann, H.-O. Ross, and M. Stynes, Numerical
Treatment of Partial Differ-
ential Equations, Springer, Berlin, 2005.
[3] D. Kinkaid and W. Cheney, Numerical Analysis.
Mathematics of Scientific Comput-
ing, Third Edition, Brooks/Cole, 2002.
[4] S. Larsen and V. Thomee, Partial Differential Equations
Springer-Verlag, Texts in Applied Mathematics 45, 2003.

in Applied Mathematics 13, Springer-Verlag, 1993.
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law1541X_case4_471-479.indd 471 12/23/15 09:21 AM
C A S E F O U R
Sustainability at Holland
America Line
By Murray Silverman, San Francisco State University. This is
an edited version of a longer case, “Protecting Our Oceans:
Sustainability at Holland America Line,” copyright © 2012 by
Murray Silverman; all rights reserved. This version was edited,
abridged, and used by permission of the author. A full set of
footnotes is available in the longer case. The case was devel-
oped with the cooperation of Holland America and the support
of the Center for Ethical and Sustainable Business at San
Francisco State University and the Campbell Foundation. This

case was prepared as a basis for class discussion rather than
to illustrate the effective or ineffective handling of an
administrative situation.
Holland America Line (HAL) was proud of its reputation as a
sustainability leader in the
global cruise industry. Bill Morani, vice president for safety
and environmental systems,
was responsible for ensuring that the company and fleet
complied with both safety and
environmental regulations and policies. In light of the maritime
industry’s significant envi-
ronmental impacts and its complex and rapidly evolving
regulatory environment, Morani
was thinking about how to prioritize the company’s current
sustainability initiatives. His
musings were interrupted as Dan Grausz, executive vice
president for fleet operations, came
into his office waving an article. The Stena Line, a ferry
operator, had reduced fuel use on
one of its vessels by installing two wind turbines on deck, the
article reported. Grausz, who
also served as leader of the company’s fuel conservation
committee, reminded Morani that
wind turbines were one of 56 initiatives HAL was evaluating. It

had been assigned a low
priority, but Grausz asked Morani if he thought that should be
reconsidered.
HAL, headquartered in Seattle, Washington, was founded as a
shipping and passenger
line in 1873 and offered its first vacation cruises in 1895. In
1989, HAL became a wholly
owned subsidiary of Carnival Corporation. HAL maintained its
own identity, operating its
own fleet, and managing its marketing, sales, and administrative
support. In 2011, HAL
operated 15 mid-size ships, mostly in the premium segment, and
expected to carry 750,000
passengers to 350 ports in 100 countries. The company had
more than 14,000 employees.
HAL was widely recognized as a leader in the cruise industry in
its environmental sus-
tainability. In 2006, HAL had received the Green Planet Award,
which recognized eco-
minded hotels, resorts, and cruise lines for outstanding
environmental standards. This
award was based on the company’s ISO14001 certification and
the installation of shore

power plug-in systems on three ships. In 2008, Virgin Holidays
awarded HAL its Respon-
sible Tourism Award based on its reduction of dockside
emissions, increased use of recy-
cling, and adoption of a training program to avoid whale strikes.
In 2011, HAL was named
the World’s Leading Green Cruise Line at the World Travel
Awards in London, and in both
2010 and 2012 the company had received the Gold
Environmental Protection Award from
the U.S. Coast Guard.
Morani was particularly proud of the progress HAL had made in
improving fuel effi-
ciency; the company had reduced its fuel use per passenger per
nautical mile by 20 percent
between 2005 and 2011. Burning less fuel meant lower
emissions of carbon dioxide, sulfur
Final PDF to printer
law1541X_case4_471-479.indd 472 12/23/15 09:21 AM

472 Cases in Business and Society
and nitrous oxides (SOX and NOX), and particulate matter
(PM). These emissions were
increasingly regulated, because of rising concerns about both
their health and environmen-
tal impacts. According to Morani:
Fuel conservation is our go-to strategy. It is a win–win. By
consuming less fuel, we
are not emitting as much exhaust containing greenhouse gases
and other pollutants,
while reducing HAL’s fuel costs. And, by the way, the money
saved through fuel
conservation can help offset the increased cost of cleaner fuel.
Morani put aside his thinking about broader sustainability
priorities in order to look into
the wind turbine idea.
The Global Cruise Industry
Taking a cruise was very popular among tourists, and the cruise
industry was one of the
fastest growing sectors of the tourism industry. The modern

cruise industry traced its
beginnings to the early 1970s, when the industry began offering
Caribbean cruises from
Miami, Florida. As it evolved, the industry created a reasonably
priced opportunity for
many people to experience a resort-type vacation. Sometimes,
cruise ships were referred
to as floating hotels or marine resorts, because they had
sleeping rooms, restaurants, enter-
tainment, shops, spas, business centers, casinos, swimming
pools, and other amenities, just
like land-based resorts.
By the mid-2010s, cruise ships traveled in every ocean,
frequently visiting the most
pristine coastal waters and sensitive marine ecosystems. Among
the most popular destina-
tions were the Caribbean, the Mediterranean Sea, various
European ports, the Bahamas,
and Alaska. Worldwide, approximately 2,000 ports were capable
of receiving cruise ships.
Destinations varied from tropical beaches like Cozumel, to
nature-based destinations such
as Alaska, to historical and culturally rich locations such as
Istanbul. The cruise product

was highly diversified, based on destination, ship design, on-
board and on-shore activi-
ties, themes, and cruise lengths; accommodations and amenities
were priced accordingly.
Classifications ranged from budget to conventional to premium
and, lastly, to luxury. The
passenger capacity of cruise ships tended to be larger in the
budget and conventional cate-
gories and varied from a few hundred to over 5,000 passengers.
The popularity of cruising was reflected in its growth. Since
1980, the number of pas-
sengers had grown by an annual rate of 7.6 percent. Between
1990 and 2010, more than
191 million passengers took a cruise. Twenty-four percent of
the American population
had cruised at least once. Passengers were predominately
Caucasian (93%), well educated,
and married (83%). Their average age was 46, with an average
household income between
$90,000 and $100,000. The leading factors in the customer’s
selection of a cruise pack-
age were destination and price; industry executives believed
that few customers consid-
ered a cruise line’s environmental practices in their choice. As

demand grew, the industry
responded by building more cruise ships. As of 2012, 256 cruise
ships plied global waters.
Newer ships tended to be bigger, and they often included
innovative amenities such as
planetariums and bowling alleys.
Cruise lines were a $30 million a year global industry. In 2012,
three major companies
dominated the industry: Carnival Corporation (52 percent of
passengers), Royal Caribbean
Cruises Ltd. (21 percent), and Norwegian Cruise Line (7
percent). Each of these compa-
nies had a number of brands, allowing them to operate within
various pricing segments.
The industry was organized into the Cruise Line Industry
Association (CLIA), whose
membership included 22 of the world’s largest cruise line
companies and accounted for
97% of the demand for cruises.

law1541X_case4_471-479.indd 473 12/23/15 09:21 AM
Case 4 Sustainability at Holland America Line 473
The World’s Oceans
HAL and the cruise industry as a whole relied on the oceans as
their most important
resource. The unspoiled waters and coral reefs at port
destinations were a major attraction
for passengers. Oceans, which covered 71 percent of the Earth’s
surface, provided many
benefits for society. They were a source of food, in the form of
fish and shellfish, and were
used for transportation and recreation, such as swimming,
sailing, diving, and surfing.
They provided biomedical organisms that helped fight disease.
And, the ocean played a
significant role in regulating the planet’s climate by absorbing
carbon dioxide and heat.
Yet, in the mid-2010s, the oceans faced many environmental
threats:
Overfishing: More than half the world’s population depended on
the oceans for their

primary source of food, yet most of the world’s fisheries were
being fished at levels above
their maximum sustainable yield. Furthermore, harmful fishing
methods were unnecessar-
ily killing turtles, dolphins, and other animals and destroying
critical habitat.
Pollution: Eighty percent of all pollution in seas and oceans
came from land-based
activities. More oil reached the ocean each year as a result of
leaking automobiles and
other nonpoint sources, for example, than was spilled by the
Exxon Valdez. An enormous
amount of oil had been accidentally spilled from ships,
destroying aquatic plant and animal
life.
Eutrophication: Another serious ocean threat was algal blooms
caused mainly by fertil-
izer and topsoil runoff and sewage discharges in coastal areas.
As algae died and decom-
posed, water was depleted of available oxygen, causing the
death of other organisms such
as fish.

Ocean acidification: Carbon dioxide in the atmosphere, caused
mainly by the burning
of fossil fuels, was a well-known contributor to global warming.
But, it also acidified the
oceans. When absorbed in water, carbon dioxide was converted
into carbonic acid, which
in turn dissolved reefs needed by organisms such as corals and
oysters, threatening their
survival.
Ocean warming: Atmospheric warming also increased the
temperature of the ocean,
reducing the generation of plankton, the base of the ocean’s
food web, and leading to sig-
nificant marine ecosystem change.
Tourism: While tourism generated vast amounts of income for
host countries, it could
also have adverse environmental impacts, especially in heavily
visited coastal areas. Sewage
and waste from resorts, hotels, and restaurants could find their
way into bays and oceans.
Careless diving, snorkeling and other tour activities could
damage coral reefs.

Environmental Impacts of the Cruise Industry
In a number of ways, the cruise industry contributed to these
threats to ocean health. The
primary inputs for a cruise were food, packaging materials,
fresh and sea water, and fuel.
As these inputs were processed over the course of a cruise, they
produced discharges or
waste with environmental impacts on water, air, and land. These
impacts are diagrammed
in Exhibit A.
Discharges to Water
The primary discharges to water from a cruise ship were
blackwater (sewage), graywa-
ter (from showers, sinks, laundry, and the galley), and bilge
water (potentially oily water
leaked from engines and equipment that accumulated in the
bilges).

474
law1541X_case4_471-479.indd 474 12/23/15 09:21 AM
Environmental Aspects and Potential Impacts
from Cruise Ship OperationsExhibit A
AIR
Air pollution, climate change, ozone layer depletion
POTENTIAL IMPACTS
Enginerboiler/Incinerator emissions
INPUTS OUTPUTS
POTENTIAL IMPACTS
WATER
Oil spills, water pollution, biodiversity imbalance
OUTPUTS
POTENTIAL

IMPACTS
Soild waste
Soil &
groundwater
pollution
Recyclables
Hazardous
waste
POTENTIAL
IMPACTS
LA
N
D
LA
N
D

Natural
resource
depletion
Food
Packaging
material
Fresh water
Fuel energy
Sea water
Ballast
water
Bridge
water
Partially
treated
organic
waste
Treated
blackwater

Graywater Permeate
Refrigerant releases
DISCHARGES
Source: Holland America Lines.
Blackwater contained pathogens, including fecal coliform
bacteria, which could con-
taminate fisheries and shellfish beds, risking human health. On
most cruise ships, sewage
was treated using a marine sanitation device (MSD) that
disinfected the waste prior to dis-
charge. A newer technology, called advanced wastewater
purification systems (AWWPS),
was capable of producing water effluent as clean as or cleaner
than that produced by many
municipal treatment plants. International and U.S. regulations
required that treated sewage
be discharged at least 3 nautical miles from shore and untreated
sewage at least 12 nautical
miles from shore. All discharges were banned in certain
sensitive zones.

Graywater could also contain pollutants, including oil,
detergents, grease, suspended
solids, nutrients, food waste, and small concentrations of
coliform bacteria. U.S. regula-
tions prohibited the discharge of graywater within three miles of
the coast in California
and Alaska. Voluntary industry standards specified a distance of
at least four miles from
the coast.
Bilge water. Regulators required that discharged bilge water
contain less than 15 ppm
(parts per million) of oil and could only be discharged while the
vessel was en route and
not operating in protected zones.
Solid and Hazardous Waste
Cruise ship waste streams could be either hazardous (e.g.,
chemicals from dry cleaning
or photo processing, solvents, and paint waste) or nonhazardous
(e.g., food waste, paper,
plastic, and glass). Waste could be discarded either in the water
or on land.

The potential impact from pollution by solid waste on the open
ocean and the coastal
environment could be significant, including aesthetic
degradation of surface waters and
law1541X_case4_471-479.indd 475 12/23/15 09:21 AM
coastal areas. Sea birds, fish, turtles, and cetaceans could be
entangled in waste and injured
or killed. The disposal of food wastes in restricted areas could
cause pollution.
Air Emissions
Cruise ship engines were designed to generate the energy they
needed both for propulsion
and for operating lights, refrigeration, heating and cooling, and
other onboard services.
The main fuel used by cruise ships was the relatively dirty-
burning heavy fuel oil (HFO).

Distillate and low-sulfur fuel oil (LSFO) offered a cleaner
alternative to HFO, but usually
cost between 10 and 50 percent more. Fuel costs typically
accounted for around 15 percent
of operating costs on a cruise ship.
Engine exhaust was the primary source of air emissions; these
included carbon dioxide,
nitrous and sulfur oxides, and particulate matter. Around 2 to 3
percent of global car-
bon dioxide emissions came from maritime shipping, mostly
from the 50,000 merchant
ships plying the ocean. The 350 cruise ships contributed in a
small way to this problem.
The impact of shipping on SOX and NOX was greater: the
maritime industry as a whole
accounted for approximately 4 percent and 7 percent,
respectively, of global SOX and
NOX emissions, with cruise ships contributing part of this.
Regulation of the Maritime Industry
Regulations governing the maritime industry and its
environmental impacts were complex
and multilayered. Shipping activities were governed by a

mixture of United Nations con-
ventions, the international law of the sea, the laws of various
nations, and voluntary rules
established by industry trade associations.
Several formal institutions and instruments provided
mechanisms for cooperation among
national governments in managing the ocean commons. The
International Maritime Organi-
zation (IMO), a specialized agency of the United Nations,
regulated the international shipping
industry. One of its most important initiatives was the IMO
Convention for the Prevention for
Pollution from Ships, known as MARPOL (for “marine
pollution”). Ships operating under
the flags of countries that had signed the MARPOL convention
were subject to its rules.
(Countries responsible for 99 percent of marine shipping had
signed.) Other international
agreements included the UN Convention on the Law of the Sea
(UNCLOS), a comprehen-
sive treaty establishing protocols for the use and exploitation of
the ocean and its resources.
The International Whaling Convention regulated the hunting of
great whales. Overall, regu-

lations of the maritime industry had become stricter over time,
as concern about the ecolog-
ical health of the oceans had grown. For example, international
bodies had created special
emission control areas, where discharges of airborne pollutants
were sharply curtailed.
The country where a ship was registered, called the flag state,
was obligated to ensure
that its ships complied with regulations set down in
international conventions to which the
flag state was a signatory. Even if a ship was registered in a
flag state that had not ratified
a particular IMO convention, it had to obey rules adopted by
any nations it visited. Since
almost all cruise ship ports were in nations that had ratified the
IMO regulations, as a prac-
tical matter, cruise ships were required to abide by IMO
regulations.
Individual nations had also established their own regulations,
and cruise ships had
to follow the rules of any country they visited. For example, in
2009 the United States
and Canada joined together to establish an Emissions Control

Area covering all of North
America, with the goal of reducing pollution in coastal waters.
In situations where national
rules were stricter than those of international conventions, the
cruise industry had to follow
the national rules.
law1541X_case4_471-479.indd 476 12/23/15 09:21 AM
476 Cases in Business and Society
In addition, the CLIA had developed its own waste management
practices and proce-
dures. In many instances, these voluntary environmental
standards exceeded the require-
ments of both U.S. and international laws. For example, while
regulations permitted the
discharge of untreated blackwater 12 nautical miles from shore,
CLIA standards called
for treating all blackwater using advanced water purification
systems, no matter how

far from shore it was discharged. However, CLIA did not
proscribe the manner in
which the voluntary standards were to implemented, nor impose
penalties for failing to
follow them.
Holland America Lines was committed to meeting or exceeding
the standards estab-
lished by all relevant international and national laws (including
those of the Netherlands,
where its ships were registered), as well as the CLIA standards.
HAL’s Sustainability Practices
HAL operated its sustainability programs relatively
independently of its parent firm,
Carnival. The Safety and Environmental Management Systems
(SEMS) Department over-
saw HAL’s programs in this area. Bill Morani served as vice
president for SEMS; he,
in turn, reported to Dan Grausz, executive vice president of
fleet operations. SEMS was
responsible for ensuring that all employees understood their
roles and responsibilities. It
also developed written environmental procedures, emergency
preparedness plans, and per-

formance targets and oversaw a rigorous environmental audit
program. Onboard each ship,
a safety, environmental and health (SHE) officer advised the
captain’s staff on compliance
policies, processes, and environmental regulatory requirements.
In 2009, HAL released its first sustainability report covering
activities from 2007–09;
a second report was issued in 2012. Their sustainability reports
used the Global Reporting
Initiative’s (GRI) G3 Guidelines as its organizing framework.
The data in this baseline
report was not independently verified, although this was not
unusual among first-time GRI
reporters. Their environmental management system (EMS) was
recertified in 2009 and
2012 as meeting ISO 14001 environmental standards.
HAL’s sustainability reports documented the company’s
progress in a number of areas.
These included the following highlights:
∙ HAL was instrumental in developing advanced wastewater
purification systems
(AWWPS) technology for use in cruise ships, first installed on

the MS Statendam in
2002. These systems used a combination of screening,
maceration, biodigestion, ultra-
filtration, and ultraviolet light to clean wastewater to a much
higher standard that con-
ventional systems. By 2012, 12 of HAL’s 15 ships used
AWWPSs (compared with 40
percent in the rest of the industry). HAL was also a leader in
improving bilge water
treatment prior to overboard discharge.
∙ HAL also had used various conservation strategies to reduce
the amount of water used
and discharged. In 2009, HAL used their environmental
management system (EMS) to
set a target of using 7 percent less water than in 2008. They
exceeded the target using a
number of approaches including low-flush toilets, low-flow
showerheads and faucets,
and specialized pool filters.
∙ HAL had taken steps to reduce its solid waste flow. Onboard,
paper and cardboard were
shredded and often incinerated to reduce the fire load carried by
the vessel. Food waste

was run through a pulper and discharged more than 12 nautical
miles from shore. The
company recycled much of its glass, paper, cardboard,
aluminum, steel cans, and plas-
tics on shore. It replaced highly toxic dry-cleaning fluids with a
nontoxic technology,
developed a paint and thinner recycling program, and
implemented a list of approved
law1541X_case4_471-479.indd 477 12/23/15 09:21 AM
chemicals to reduce the use of toxics. HAL donated many
partially used products and
reusable items (mattresses, toiletries, linen, clothing, etc.) to
nonprofit organizations.
One supply issue that received special attention was the
sustainability of seafood served
on board. In 2010, Hal partnered with the Marine Conservation

Institute (MCI) to protect
marine ecosystems in a program called “Our Marvelous
Oceans.” MCI was a nonprofit
organization working with scientists, politicians, government
officials, and other organi-
zations around the world to protect essential ocean places and
the wild species in them.
Under the terms of the partnership, HAL committed to
purchasing sustainable seafood to
be served on board. It also developed a series of video programs
about the oceans to be
shown to guests, and supported MCI grants to graduate students
and young scientists in
marine ecology. As part of the partnership, MCI staff evaluated
the sustainability of over
40 species of fish. HAL committed to use best choice items
where available and to discon-
tinue purchase of not-sustainable species. When more
information was needed, HAL went
back to the suppliers, who in many cases were able to find
sustainable alternatives (such as
Dover sole caught with hook and lines). HAL’s senior managers
embraced this program,
even though in some cases the cost of fish was higher.

Managing Fuel Conservation at HAL
As part of its overall sustainability initiatives, in 2005, HAL’s
parent, Carnival Corpo-
ration, set an ambitious goal of increasing fuel efficiency as
measured by the amount of
fuel used per lower berth per nautical mile by 20 percent by
2015. In order to meet this
goal, HAL had established a cross-functional fuel conservation
committee in 2007 that
systematically identified and assessed fuel reduction
opportunities, based primarily on
projected fuel savings and return on investment (ROI). The
committee had been very
effective in adopting successful initiatives based on established
financial criteria, and
HAL reached the 2015 target in 2011. Exhibit B shows the
company’s improved fuel
efficiency over time, as well as its mix of fuels used. It shows
that although fuel use
increased overall (due to an expanding fleet and more
passengers), fuel used per lower
berth steadily decreased.
HAL had reduced its fuel use through a variety of techniques,
including:

∙ Using more energy-efficient equipment and ships.
∙ Conserving energy.
∙ Plugging into shore power when docked.
∙ Encouraging competition among vessels on energy efficiency.
∙ Sharing best practices from high-performing ships.
∙ Providing monetary incentives to senior shipboard staff to
encourage fuel conservation
practices.
In 2012, the fuel conservation committee was evaluating close
to 50 initiatives to
improve efficiency even further. These initiatives fell into five
broad categories, most of
which required capital investments in new and modified
equipment:
∙ Sailing and maneuvering (6 initiatives), such as using software
to optimize speed and
maneuvering.
∙ Modifying or adding equipment (28 initiatives), such as
upgrading air conditioner
chiller control systems.

∙ Operational improvements (8 initiatives), such as running a
seawater cooling pump
while in port.
478
law1541X_case4_471-479.indd 478 12/23/15 09:21 AM
∙ Monitoring various sources of energy consumption (10
initiatives), such as installing
meters in electrical substations to monitor the energy
consumption of various users.
∙ Waste heat recovery (4 initiatives), such as adding an
additional heat exchanger to reuse
high temperature waste heat for potable water heating.
The committee’s spreadsheets included estimates of potential
savings from each initiative and
the cost per ship. Typically, the estimates of savings were

measured in terms of percentage of
overall fuel budget. For the 38 initiatives for which estimates
had been made, 13 would probably
save 0.25 percent of fuel or less, 16 would save between 0.26
and 0.99 percent, and 9 would save
more than 1 percent. The committee also tracked whether each
initiative was proven or assumed
to be viable and its stage of implementation (study, funding
required, implemented, or discon-
tinued). Finally, based on all of this information, the committee
assigned a priority (1, 2, or 3) to
each initiative. Because the capital budget available to pursue
fuel conservation projects was lim-
ited, even initiatives with a priority of 1 might not be
implemented, or might not be implemented
fleetwide. When the committee concluded that a proposed fuel
conservation initiative should be
implemented, it was pilot-tested on a single ship. Performance
was tracked, and if the results met
investment criteria, the initiative would be eligible to be rolled
out to other ships.
Because of the unproven nature of the wind turbine initiative
and skepticism on the part
of HAL’s engineering department, the fuel conservation

committee had earlier assigned
it a priority “3” and an estimated fuel savings of less than 0.25
percent. However, when
Morani read the article about Stena Line (a ferry line providing
service between Britain,
Holland, and Ireland), he wondered if this option should be
revisited. He learned that the
two turbines installed on the Stena Jutlandica could generate
about 23,000 kilowatt hours
per year, equivalent to the annual domestic electricity
consumption of four average homes
or a reduction in fuel consumption of between 80 and 90 tons
per year.
Fuel Use and Fuel Efficiency at Holland
America LineExhibit B
Source: Holland America Line.
Note: Fuel efficiency is measured as metric tons of fuel per
lower berth/nautical mile.
480,000
Fuel Use and Fuel E�ciency

Distillate Consumed in tonnes
LSFO Consumed
HFO Consumed in tonnes
Fuel E�ciency (in MT/ALB-N MT)
470,000
460,000
450,000
440,000
430,000
420,000
410,000
400,000
390,000

2007 2008 2009 2010 2011
0.000230
0.000225
0.000220
0.000215
0.000210
0.000205
0.000200
0.000195
0.000190
0.000185

law1541X_case4_471-479.indd 479 12/23/15 09:21 AM
Morani began to inquire internally at HAL about the wind
turbine idea. One of his
direct reports had received unsubstantiated information that the
Stena Line installation
was projected to be very cost effective, and—contrary to
intuition—the turbines actually
reduced aerodynamic drag on the ferry. Morani also found
another article describing how
Hornblower Cruises planned to launch a hybrid vessel to take
passengers on sightseeing,
dinner, and social events in New York Harbor. This 600-
passenger vessel would incorpo-
rate wind turbines, solar panels, and hydrogen fuel cells in
addition to its diesel engine.
The company believed the combination of alternative power
generators would result in
fuel savings that justified the investment.
Morani also consulted with Pieter Rijkaart, former director of
New Builds, who had led

the design and built most of HAL’s current fleet. Rijkaart
echoed the skepticism expressed
by other engineers. For example, the engineers had noted that a
cruise liner was much
larger and more streamlined than a ferry, raising questions
about the applicability of the
Stena Line’s performance results. Cost was also an issue. A
pilot-test on one ship would
require a large up-front investment in addition to the cost of the
turbine, as it would have
to be anchored to the deck and tied into the electrical grid on
the ship. Rijkaart also voiced
aesthetic concerns. Cruise ships were designed to be beautiful,
and having bulky wind tur-
bines on the deck could be an eyesore. Lastly, the amount of
energy supplied by the wind
turbines would account for an extremely small percentage of the
ship’s energy needs.
Morani wondered whether using wind turbines might bring
intangible benefits. HAL
had already demonstrated a proactive interest in alternative
energy initiatives. For exam-
ple, HAL had installed heat reflective film on windows to
reduce the transfer of heat to

the interior, thus reducing the load on air conditioners. At a cost
of $170,000 per ship, and
a projected fuel savings between 0.5 to1.0 percent, three ships
had already installed this
technology, and other ships awaited funding. HAL had adopted
an initiative involving the
pumping of used cooking oil into the fuel line. This low-cost
option had resulted in both
the reduction of fossil fuel and avoidance of the disposal cost of
drums of used cooking oil.
Wind turbines represented another opportunity for HAL to
explore using alternative
energy. While this could contribute to HAL’s reputation as a
sustainability leader in the
industry, Morani did not believe that reputation should be
factored into the decision. “We
don’t talk about whether something will get good press,” he
commented. While the tur-
bines would produce only a very small amount of the electricity
used on the boat, they
would contribute to reduced fuel use. Morani did not have
enough information to estimate
ROI or payback. Given the dozens of other proposed initiatives,
he wondered whether it

made sense to expend effort on this particular initiative. On the
other hand, he commented,
“I would be concerned that we could be missing an
opportunity.” Morani was eager to pull
together his thinking on the wind turbine initiative for the
upcoming fuel conservation
committee meeting.
Discussion
Questions
1. What are the most significant environmental issues facing
Holland America Line (HAL)?
2. In what ways has HAL gone “beyond compliance” in its
environmental initiatives?
3. Do you consider HAL an ecologically sustainable
organization (ESO), and why or why
not? What additional steps would HAL need to take to become
an ESO?
4. What are the advantages and disadvantages to HAL of its
sustainability practices?
5. What action would you recommend Morani take with respect
to the wind turbine initia-

tive? If you are not sure, what additional information would be
helpful?
Texts in Applied Mathematics 45
Editors
J.E. Marsden
L. Sirovich
S.S. Antman
Advisors
G. Iooss
P. Holmes
D. Barkley

M. Dellnitz
P. Newton
Texts in Applied Mathematics
1. Sirovich: Introduction to Applied Mathematics.
2. Wiggins: Introduction to Applied Nonlinear Dynamical
Systems and Chaos.
3. Hale/Koçak: Dynamics and Bifurcations.
4. Chorin/Marsden: A Mathematical Introduction to Fluid
Mechanics, Third Edition.
5. Hubbard/West: Differential Equations: A Dynamical Systems
Approach: Ordinary
Differential Equations.
6. Sontag: Mathematical Control Theory: Deterministic Finite
Dimensional Systems,

Second Edition.
7. Perko: Differential Equations and Dynamical Systems, Third
Edition.
8. Seaborn: Hypergeometric Functions and Their Applications.
9. Pipkin: A Course on Integral Equations.
10. Hoppensteadt/Peskin: Modeling and Simulation in Medicine
and the Life Sciences,
Second Edition.
11. Braun: Differential Equations and Their Applications,
Fourth Edition.
12. Stoer/Bulirsch: Introduction to Numerical Analysis, Third
Edition.
13. Renardy/Rogers: An Introduction to Partial Differential
Equations.
14. Banks: Growth and Diffusion Phenomena: Mathematical

Frameworks and
Applications.
15. Brenner/Scott: The Mathematical Theory of Finite Element
Methods, Second Edition.
16. Van de Velde: Concurrent Scientific Computing.
17. Marsden/Ratiu: Introduction to Mechanics and Symmetry,
Second Edition.
18. Hubbard/West: Differential Equations: A Dynamical
Systems Approach:
Higher-Dimensional Systems.
19. Kaplan/Glass: Understanding Nonlinear Dynamics.
20. Holmes: Introduction to Perturbation Methods.
21. Curtain/Zwart: An Introduction to Infinite-Dimensional
Linear Systems Theory.
22. Thomas: Numerical Partial Differential Equations: Finite

Difference Methods.
23. Taylor: Partial Differential Equations: Basic Theory.
24. Merkin: Introduction to the Theory of Stability.
25. Naber: Topology, Geometry, and Gauge Fields:
Foundations.
26. Polderman/Willems: Introduction to Mathematical Systems
Theory: A Behavioral
Approach.
27. Reddy: Introductory Functional Analysis: with Applications
to Boundary Value
Problems and Finite Elements.
28. Gustafson/Wilcox: Analytical and Computational Methods
of Advanced Engineering
Mathematics.
29. Tveito/Winther: Introduction to Partial Differential

Equations: A Computational
Approach.
30. Gasquet/Witomski: Fourier Analysis and Applications:
Filtering, Numerical
Computation, Wavelets.
31. Brémaud: Markov Chains: Gibbs Fields, Monte Carlo
Simulation, and Queues.
32. Durran: Numerical Methods for Wave Equations in
Geophysical Fluid Dynamics.
33. Thomas: Numerical Partial Differential Equations:
Conservation Laws and Elliptic
Equations.
(continued after index)
Stig Larsson · Vidar Thomée

Partial
Differential Equations
with Numerical
Methods
123
Stig Larsson
Vidar Thomée
Mathematical Sciences
Chalmers University of Technology
and University of Gothenburg
412 96 Göteborg
Sweden
[email protected][email protected]
Series Editors
J.E. Marsden
Control and Dynamical Systems, 107-81
California Institute of Technology

Pasadena, CA 91125
USA
[email protected]
L. Sirovich
Laboratory of Applied Mathematics
Mt. Sinai School of Medicine
Box 1012
New York City, NY 10029-6574
USA
[email protected]
S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA
[email protected]
First softcover printing 2009
ISBN 978-3-540-88705-8 e-ISBN 978-3-540-88706-5
DOI 10.1007/978-3-540-88706-5

Texts in Applied Mathematics ISSN 0939-2475
Library of Congress Control Number: 2008940064
Mathematics Subject Classification (2000): 35-01, 65-01
c∇ 2009, 2003 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved,
whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting,
reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in
data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the
German Copyright Law of September 9,
1965, in its current version, and permission for use must always
be obtained from Springer. Violations
are liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names,
trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are
exempt from the relevant protective laws
and regulations and therefore free for general use.

Coverdesign: WMXDesign GmbH, Heidelberg
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Series Preface
Mathematics is playing an ever more important role in the
physical and
biological sciences, provoking a blurring of boundaries between
scientific
disciplines and a resurgence of interest in the modern as well as
the classical
techniques of applied mathematics. This renewal of interest,
both in re-
search and teaching, has led to the establishment of the series
Texts in
Applied Mathematics (TAM).
The development of new courses is a natural consequence of a

high level
of excitement on the research frontier as newer techniques, such
as numeri-
cal and symbolic computer systems, dynamical systems, and
chaos, mix
with and reinforce the traditional methods of applied
mathematics. Thus,
the purpose of this textbook series is to meet the current and
future needs
of these advances and to encourage the teaching of new courses.
TAM will publish textbooks suitable for use in advanced
undergraduate
and beginning graduate courses, and will complement the
Applied Mathe-
matical Sciences (AMS) series, which will focus on advanced
textbooks and
research-level monographs.
Pasadena, California J.E. Marsden
New York, New York L. Sirovich
College Park, Maryland S.S. Antman

Preface
Our purpose in this book is to give an elementary, relatively
short, and hope-
fully readable account of the basic types of linear partial
differential equations
and their properties, together with the most commonly used
methods for their
numerical solution. Our approach is to integrate the
mathematical analysis
of the differential equations with the corresponding numerical
analysis. For
the mathematician interested in partial differential equations or
the person
using such equations in the modelling of physical problems, it is
important
to realize that numerical methods are normally needed to find
actual values
of the solutions, and for the numerical analyst it is essential to
be aware that
numerical methods can only be designed, analyzed, and
understood with suf-
ficient knowledge of the theory of the differential equations,
using discrete

analogues of properties of these.
In our presentation we study the three major types of linear
partial differ-
ential equations, namely elliptic, parabolic, and hyperbolic
equations, and for
each of these types of equations the text contains three chapters.
In the first
of these we introduce basic mathematical properties of the
differential equa-
tion, and discuss existence, uniqueness, stability, and regularity
of solutions
of the various boundary value problems, and the remaining two
chapters are
devoted to the most important and widely used classes of
numerical methods,
namely finite difference methods and finite element methods.
Historically, finite difference methods were the first to be
developed and
applied. These are normally defined by looking for an
approximate solution
on a uniform mesh of points and by replacing the derivatives in
the differential
equation by difference quotients at the mesh-points. Finite

element methods
are based instead on variational formulations of the differential
equations and
determine approximate solutions that are piecewise polynomials
on some par-
tition of the domain under consideration. The former method is
somewhat
restricted by the difficulty of adapting the mesh to a general
domain whereas
the latter is more naturally suited for a general geometry. Finite
element
methods have become most popular for elliptic and also for
parabolic prob-
lems, whereas for hyperbolic equations the finite difference
method continues
to dominate. In spite of the somewhat different philosophy
underlying the
two classes it is more reasonable in our view to consider the
latter as further
Preface
developments of the former rather than as competitors, and we

feel that the
practitioner of differential equations should be familiar with
both.
To make the presentation more easily accessible, the elliptic
chapters are
preceded by a chapter about the two-point boundary value
problem for a
second order ordinary differential equation, and those on
parabolic and hy-
perbolic evolution equations by a short chapter about the initial
value prob-
lem for a system of ordinary differential equations. We also
include a chapter
about eigenvalue problems and eigenfunction expansion, which
is an impor-
tant tool in the analysis of partial differential equations. There
we also give
some simple examples of numerical solution of eigenvalue
problems.
The last chapter provides a short survey of other classes of
numerical
methods of importance, namely collocation methods, finite
volume methods,

spectral methods, and boundary element methods.
The presentation does not presume a deep knowledge of
mathematical and
functional analysis. In an appendix we collect some of the basic
material that
we need in these areas, mostly without proofs, such as elements
of abstract
linear spaces and function spaces, in particular Sobolev spaces,
together with
basic facts about Fourier transforms. In the implementation of
numerical
methods it will normally be necessary to solve large systems of
linear algebraic
equations, and these generally have to be solved by iterative
methods. In a
second appendix we therefore include an orientation about such
methods.
Our purpose has thus been to cover a rather wide variety of
topics, notions,
and ideas, rather than to expound on the most general and far-
reaching
results or to go deeply into any one type of application. In the
problem

sections, which end the various chapters, we sometimes ask the
reader to
prove some results which are only stated in the text, and also to
further
develop some of the ideas presented. In some problems we
propose testing
some of the numerical methods on the computer, assuming that
Matlab or
some similar software is available. At the end of the book we
list a number
of standard references where more material and more detail can
be found,
including issues concerned with implementation of the
numerical methods.
This book has developed from courses that we have given over a
rather
long period of time at Chalmers University of Technology and
Göteborg Uni-
versity originally for third year engineering students but later
also in begin-
ning graduate courses for applied mathematics students. We
would like to
thank the many students in these courses for the opportunities
for us to test

our ideas.
Göteborg, Stig Larsson
January, 2003 Vidar Thomée
In the second printing 2005 we have corrected several misprints
and minor
inadequacies, and added a few problems. SL & VT
VIII
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1
1.2 Notation and Mathematical Preliminaries . . . . . . . . . . . . . .
. . . . 4
1.3 Physical Derivation of the Heat Equation . . . . . . . . . . . . . .
. . . . 7
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 12

2 A Two-Point Boundary Value Problem . . . . . . . . . . . . . . . . .
. . 15
2.1 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 15
2.2 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 18
2.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 20
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 23
3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 25
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 25
3.2 A Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 26
3.3 Dirichlet’s Problem for a Disc. Poisson’s Integral . . . . . . . .
. . . 28
3.4 Fundamental

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