Because of the inverse monotonicity and from (1.32) (5) the vectors pos-

(i)

tulated in Theorem 1.14 have to satisfy wh ≥ 0 necessarily for i = 1, 2.

Thus stability with respect to · ∞ of the method de¬ned by (1.31) as-

suming (1.32) (1)“(3), (4)* is guaranteed if a vector 0 ¤ wh ∈ RM1 and a

constant C > 0 independent of h can be found such that

Ah wh ≥ 1 and |w h |∞ ¤ C . (1.41)

Finally, this will be proven for the ¬ve-point stencil discretization (1.1),

(1.2) on the rectangle „¦ = (0, a) — (0, b) for C = 16 (a2 + b2 ).

1

For this reason we de¬ne polynomials of second degree w1 , w2 by

1 1

x(a ’ x) y(b ’ y) .

w1 (x) := and w2 (y) := (1.42)

4 4

It is clear that w1 (x) ≥ 0 for all x ∈ [0, a] and w2 (y) ≥ 0 for all y ∈ [0, b].

Furthermore, we have w1 (0) = 0 = w1 (a) and w2 (0) = 0 = w2 (b), and

1 1

w1 (x) = ’and w2 (y) = ’ .

2 2

Therefore w1 and w2 are strictly concave and attain their maximum in a 2

b

and 2 , respectively. Thus the function w(x, y) := w1 (x) + w2 (x) satis¬es

’∆w =1 in „¦ ,

(1.43)

≥0

w on ‚„¦ .

Now let w h ∈ RM1 be, for a ¬xed ordering, the representation of the grid

function wh de¬ned by

for i = 1, . . . , l ’ 1 , j = 1, . . . , m ’ 1 .

(wh )(ih, jh) := w(ih, jh)

1.4. Maximum Principles and Stability 43

Analogously, let w h ∈ RM2 be the representation of the function wh de-

ˆ ˆ

—

¬ned on ‚„¦h . As can be seen from the error representation in Lemma 1.2,

statement 4, the di¬erence quotient ‚ ’ ‚ + u(x) is exact for polynomials of

second degree. Therefore, we conclude from (1.43) that

Ah w h = ’Ah w h + 1 ≥ 1 ,

ˆˆ

which ¬nally implies

a b 12

|w h |∞ = wh ¤w (a + b2 ) .

= w1 + w2 =

∞ ∞

2 2 16

This example motivates the following general procedure to construct wh ∈

RM1 and a constant C such that (1.41) is ful¬lled.

Assume that the boundary value problem under consideration reads in

an abstract form

x ∈ „¦,

(Lu)(x) = f (x) for

(1.44)

x ∈ ‚„¦ .

(Ru)(x) = g(x) for

Similar to (1.43) we can consider ” in case of existence ” a solution w

of (1.44) for some f, g, such that f (x) ≥ 1 for all x ∈ „¦, g(x) ≥ 0 for all

x ∈ „¦. If w is bounded on „¦, then

(wh )i := w(xi ), i = 1, . . . , M1 ,

for the (non-Dirichlet) grid points xi , is a candidate for wh . Obviously,

|w h |∞ ¤ w .

∞

Correspondingly, we set

(w h )i = w(xi ) ≥ 0 ,

ˆ i = M1 + 1, . . . , M2 ,

for the Dirichlet-boundary grid points.

The exact ful¬llment of the discrete equations by wh cannot be expected

anymore, but in case of consistency the residual can be made arbitrarily

small for small h. This leads to

Theorem 1.15 Assume that a solution w ∈ C(„¦) of (1.44) exists for data

f ≥ 1 and g ≥ 0. If the discretization of the form (1.31) is consistent with

˜

(1.44) (for these data), and there exists H > 0 so that for some ± > 0 :

˜

’Ah wh + f ≥ ±1 h¤H ,

ˆˆ ˜

˜ for (1.45)

then for every 0 < ± < ± there exists H > 0, so that

˜

Ah w h ≥ ±1 h¤H.

for

Proof: Set

„ h := Ah w h + Ah w h ’ f

ˆˆ

44 1. Finite Di¬erence Method for the Poisson Equation

for the consistency error, then

|„ h |∞ ’ 0 for h ’ 0 .

Thus

„ h ’ Ah w h + f

ˆˆ

Ah wh =

≥ ’|„ h |∞ 1 + ±1 for h ¤ H

˜

˜

≥ h¤H

±1 for

2

and some appropriate H > 0.

Thus a proper choice in (1.41) is

1 1

wh and C := w ∞. (1.46)

± ±

The condition (1.45) is not critical: In case of Dirichlet boundary conditions

and (1.32) (5) (for corresponding rows i of Ah ) then, due to (f )i ≥ 1, we

ˆ

can even choose ± = 1. The discussion of Neumann boundary conditions

˜

following (1.24) shows that the same can be expected.

Theorem 1.15 shows that for a discretization with an inverse monotone

system matrix consistency already implies stability.

To conclude this section let us discuss the various ingredients of (1.32)

or (1.32)* that are su¬cient for a range of properties from the inverse

monotonicity up to a strong maximum principle: For the ¬ve-point stencil

on a rectangle all the properties are valid for Dirichlet boundary conditions.

If partly Neumann boundary conditions appear, the situation is the same,

but now close and far from the boundary refers to its Dirichlet part. In

the interpretation of the implications one has to take into account that the

heterogeneities of the Neumann boundary condition are now part of the

right-hand side f , as seen, e.g., in (1.26). If mixed boundary conditions are

applied, as

‚ν u + ±u = g on “2 (1.47)

for some “2 ‚ “ and ± = ±(x) > 0, then the situation is the same again

if ±u is approximated just by evaluation, at the cost that (4)* no longer

holds. The situation is similar if reaction terms appear in the di¬erential

equation (see Exercise 1.10).

Exercises

1.11 Give an example of a matrix Ah ∈ RM1 ,M2 that can be used in the

ˆ

proof of Theorem 1.12.

1.12 Show that the transposition of an M-matrix is again an M-matrix.

Exercises 45

1.13 In the assumptions of Theorem 1.9 substitute (1.32) (4) by (4)* and

amend (6) to

#

(6) Condition (1.32) (6) is valid and

M1

(Ah )rs > 0 ’ there exists s ∈ {M1 , . . . , M } such that (Ah )rs < 0.

ˆ

s=1

Under these conditions prove a weak maximum principle as in Theorem 1.9.

1.14 Assuming the existence of wh ∈ RM1 such that Ah wh ≥ 1 and

|wh |∞ ¤ C for some constant C independent of h, show directly (without

Theorem 1.14) a re¬ned order of convergence estimate on the basis of an

order of consistency estimate in which also the shape of wh appears.

2

The Finite Element Method

for the Poisson Equation

The ¬nite element method, frequently abbreviated by FEM, was devel-

oped in the ¬fties in the aircraft industry, after the concept had been

independently outlined by mathematicians at an earlier time. Even today

the notions used re¬‚ect that one origin of the development lies structural

mechanics. Shortly after this beginning, the ¬nite element method was ap-

plied to problems of heat conduction and ¬‚uid mechanics, which form the

application background of this book.

An intensive mathematical analysis and further development was started

in the later sixties. The basics of this mathematical description and analy-

sis are to be developed in this and the following chapter. The homogeneous

Dirichlet boundary value problem for the Poisson equation forms the

paradigm of this chapter, but more generally valid considerations will be

emphasized. In this way the abstract foundation for the treatment of more

general problems in Chapter 3 is provided. In spite of the importance of the

¬nite element method for structural mechanics, the treatment of the linear

elasticity equations will be omitted. But we note that only a small expense

is necessary for the application of the considerations to these equations.

We refer to [11], where this is realized with a very similar notation.

2.1 Variational Formulation for the Model Problem

We will develop a new solution concept for the boundary value problem

(1.1), (1.2) as a theoretical foundation for the ¬nite element method. For

2.1. Variational Formulation 47

such a solution, the validity of the di¬erential equation (1.1) is no longer re-

quired pointwise but in the sense of some integral average with “arbitrary”

weighting functions •. In the same way, the boundary condition (1.2) will

be weakened by the renunciation of its pointwise validity.

For the present, we want to con¬ne the considerations to the case of

homogeneous boundary conditions (i.e., g ≡ 0), and so we consider the

following homogeneous Dirichlet problem for the Poisson equation: Given

a function f : „¦ ’ R, ¬nd a function u : „¦ ’ R such that

’∆u = f in „¦ , (2.1)

u= 0 on ‚„¦ . (2.2)

In the following let „¦ be a domain such that the integral theorem of

Gauss is valid, i.e. for any vector ¬eld q : „¦ ’ Rd with components in

C(„¦) © C 1 („¦) it holds

∇ · q(x) dx = ν(x) · q(x) dσ . (2.3)

„¦ ‚„¦

Let the function u : „¦ ’ R be a classical solution of (2.1), (2.2) in the

sense of De¬nition 1.1, which additionally satis¬es u ∈ C 1 („¦) to facili-

∞

tate the reasoning. Next we consider arbitrary v ∈ C0 („¦) as so-called test

functions. The smoothness of these functions allows all operations of di¬er-

∞

entiation, and furthermore, all derivatives of a function v ∈ C0 („¦) vanish

on the boundary ‚„¦. We multiply equation (2.1) by v, integrate the result

over „¦, and obtain

f (x)v(x) dx = ’ ∇ · (∇u)(x) v(x) dx

f, v =

0

„¦ „¦

∇u(x) · ∇v(x) dx ’ ∇u(x) · ν(x) v(x) dσ

= (2.4)

„¦ ‚„¦

∇u(x) · ∇v(x) dx .

=

„¦

The equality sign at the beginning of the second line of (2.4) is obtained

by integration by parts using the integral theorem of Gauss with q = v∇u .

The boundary integral vanishes because v = 0 holds on ‚„¦.

∞

If we de¬ne, for u ∈ C 1 („¦), v ∈ C0 („¦), a real-valued mapping a by

∇u(x) · ∇v(x) dx ,

a(u, v) :=

„¦

then the classical solution of the boundary value problem satis¬es the

identity

∞

for all v ∈ C0 („¦) .

a(u, v) = f, v (2.5)

0

48 2. Finite Element Method for Poisson Equation

∞

The mapping a de¬nes a scalar product on C0 („¦) that induces the norm

1/2

|∇u| dx

2

u := a(u, u) = (2.6)

a

„¦

(see Appendix A.4 for these notions). Most of the properties of a

scalar product are obvious. Only the de¬niteness (A4.7) requires further

considerations. Namely, we have to show that

(∇u · ∇u) (x) dx = 0 ⇐’ u ≡ 0 .

a(u, u) =

„¦

To prove this assertion, ¬rst we show that a(u, u) = 0 implies ∇u(x) = 0

for all x ∈ „¦. To do this, we suppose that there exists some point x ∈ „¦

¯

such that ∇u(¯) = 0. Then (∇u · ∇u) (¯) = |∇u| (¯) > 0. Because of

2

x x x

the continuity of ∇u, a small neighbourhood G of x exists with a positive

¯

measure |G| and |∇u|(x) ≥ ± > 0 for all x ∈ G. Since |∇u|2 (x) ≥ 0 for all

x ∈ „¦, it follows that

2

|∇u| (x) dx ≥ ±2 |G| > 0 ,

„¦

which is in contradiction to a(u, u) = 0. Consequently, ∇u(x) = 0 holds

for all x ∈ „¦; i.e., u is constant in „¦. Since u(x) = 0 for all x ∈ ‚„¦, the

assertion follows.

∞

Unfortunately, the space C0 („¦) is too small to play the part of the basic

∞

space because the solution u does not belong to C0 („¦) in general. The

identity (2.4) is to be satis¬ed for a larger class of functions, which include,

as an example for v, the solution u and the ¬nite element approximation

to u to be de¬ned later.

For the present we de¬ne as the basic space V ,

V := u : „¦ ’ R u ∈ C(„¦) , ‚i u exists and is piecewise

¯

(2.7)

continuous for all i = 1, . . . , d, u = 0 on ‚„¦ .

To say that ‚i u is piecewise continuous means that the domain „¦ can be

decomposed as follows:

¯ ¯

„¦= „¦j ,

j

with a ¬nite number of open sets „¦j , with „¦j © „¦k = … for j = k , and ‚i u

¯

is continuous on „¦j and it can continuously be extended on „¦j .

Then the following properties hold:

• a is a scalar product also on V ,

∞

• C0 („¦) ‚ V ,

∞

• C0 („¦) is dense in V with respect to · a ; i.e., for any u ∈ V (2.8)

∞

a sequence (un )n∈N in C0 („¦) exists such that un ’u a ’ 0

for n ’ ∞,

2.1. Variational Formulation 49

∞

• ·

C0 („¦) is dense in V with respect to 0. (2.9)

The ¬rst and second statements are obvious. The two others require a

certain technical e¬ort. A more general statement will be formulated in

Theorem 3.7.

With that, we obtain from (2.5) the following result:

Lemma 2.1 Let u be a classical solution of (2.1), (2.2) and let u ∈ C 1 („¦).

¯

Then

for all v ∈ V .

a(u, v) = f, v (2.10)

0

Equation (2.10) is also called a variational equation.

∞

Proof: Let v ∈ V . Then vn ∈ C0 („¦) exist with vn ’ v with respect

to · 0 and also to · a . Therefore, it follows from the continuity of the

bilinear form with respect to · a (see (A4.22)) and the continuity of the

functional de¬ned by the right-hand side v ’ f, v 0 with respect to · 0

(because of the Cauchy“Schwarz inequality in L2 („¦)) that

’ f, v a(u, vn ) ’ a(u, v) for n ’ ∞ .

f, vn and

0 0

2

Since a(u, vn ) = f, vn 0 , we get a(u, v) = f, v 0 .

The space V in the identity (2.10) can be further enlarged as long as (2.8)

and (2.9) will remain valid. This fact will be used later to give a correct

de¬nition.

De¬nition 2.2 A function u ∈ V is called a weak (or variational) solution

of (2.1), (2.2) if the following variational equation holds:

for all v ∈ V .

a(u, v) = f, v 0

If u models e.g. the displacement of a membrane, this relation is called

the principle of virtual work.

Lemma 2.1 guarantees that a classical solution u is a weak solution.

The weak formulation has the following properties:

• It requires less smoothness: ‚i u has to be only piecewise continuous.

• The validity of the boundary condition is guaranteed by the de¬nition

of the function space V .

We now show that the variational equation (2.10) has exactly the same

solution(s) as a minimization problem:

Lemma 2.3 The variational equation (2.10) has the same solutions u ∈ V

as the minimization problem

F (v) ’ min for all v ∈ V , (2.11)

50 2. Finite Element Method for Poisson Equation

where

1 1

a(v, v) ’ f, v ’ f, v

2

F (v) := = v .

a

0 0

2 2

Proof: (2.10) ’ (2.11):

Let u be a solution of (2.10) and let v ∈ V be chosen arbitrarily. We de¬ne

w := v ’ u ∈ V (because V is a vector space), i.e., v = u + w. Then, using

the bilinearity and symmetry, we have

1

a(u + w, u + w) ’ f, u + w 0

F (v) =

2

1 1

a(u, u) + a(u, w) + a(w, w) ’ f, u ’ f, w

= (2.12)

0 0

2 2

1

F (u) + a(w, w) ≥ F (u) ,

=

2

where the last inequality follows from the positivity of a; i.e., (2.11) holds.

(2.10) ⇐ (2.11):

Let u be a solution of (2.11) and let v ∈ V , µ ∈ R be chosen arbitrarily. We

de¬ne g(µ) := F (u + µv) for µ ∈ R. Then

g(µ) = F (u + µv) ≥ F (u) = g(0) for all µ ∈ R ,

because u + µv ∈ V ; i.e., g has a global minimum at µ = 0.

It follows analogously to (2.12):

µ2

1

g(µ) = a(u, u) ’ f, u 0 + µ (a(u, v) ’ f, v 0 ) + a(v, v) .

2 2

Hence the function g is a quadratic polynomial in µ, and in particular,

g ∈ C 1 (R) is valid. Therefore we obtain the necessary condition

0 = g (µ) = a(u, v) ’ f, v 0