FINITE ELEMENT METHODS FOR THE NUMERICALSOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

Vassilios A. Dougalis

Department of Mathematics, University of Athens, Greece

and

Institute of Applied and Computational Mathematics, FORTH, Greece

Revised edition 2013

PREFACE

This is the current version of notes that I have used for the past thirty-five years

in graduate courses at the University of Tennessee, Knoxville, the University of Crete,

the National Technical University of Athens, and the University of Athens. They have

evolved, and aged, over the years but I hope they may still prove useful to students

interested in learning the basic theory of Galerkin - finite element methods and some

facts about Sobolev spaces.

I first heard about approximation with cubic Hermite functions and splines from

George Fix in the Numerical Analysis graduate course at Harvard in the fall of 1971, and

also, subsequently, from Garrett Birkhoff of course. But most of the basic techniques of

the analysis of Galerkin methods I learnt from courses and seminars that Garth Baker

taught at Harvard during the period 1973-75.

Over the years I was fortunate to be associated with and learn more about Galerkin

methods from Max Gunzburger, Ohannes Karakashian, Larry Bales, Bill McKinney,

George Akrivis, Vidar Thomee and my students; for my debt to the latter it is apt to

say that ,

oo.

I would like to thank very much Dimitri Mitsoudis and Gregory Kounadis for trans-

forming the manuscript into TEX.

V. A. Dougalis

Athens, March 2012

In the revised 2013 edition, two new chapters 6 and 7 on Galerkin finite element

methods for parabolic and second-order hyperbolic equations were added. These had

previously existed in hand-written form, and I would like to thank Gregory Kounadis

and Leetha Saridakis for writing them in TEX.

V. A. Dougalis

Athens, February 2013

i

Contents

1 Some Elements of Hilbert Space Theory 1

1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Inner product, Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Some topological concepts . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Examples of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 The Projection Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Bounded (continuous) linear functionals on a Hilbert space . . . . . . . 14

1.8 Bounded (continuous) linear operators on a Hilbert space . . . . . . . . 17

1.9 The LaxMilgram and the Galerkin theorems . . . . . . . . . . . . . . 19

2 Elements of the Theory of Sobolev Spaces and Variational Formula-

tion of BoundaryValue Problems in One Dimension 26

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 The Sobolev space H1(I) . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 The Sobolev spaces Hm(I), m = 2, 3, 4, . . . . . . . . . . . . . . . . . . . 46

2.5 The space0

H1(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Twopoint boundaryvalue problems . . . . . . . . . . . . . . . . . . . 53

2.6.1 Zero Dirichlet boundary conditions. . . . . . . . . . . . . . . . . 53

2.6.2 Neumann boundary conditions. . . . . . . . . . . . . . . . . . . 58

3 Galerkin Finite Element Methods for TwoPoint BoundaryValue

Problems 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

ii

3.2 The Galerkinfinite element method with piecewise linear, continuous

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 An indefinite problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Approximation by Hermite cubic functions and cubic splines . . . . . . 84

3.4.1 Hermite, piecewise cubic functions . . . . . . . . . . . . . . . . . 84

3.4.2 Cubic splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Results from the Theory of Sobolev Spaces and the Variational For-

mulation of Elliptic BoundaryValue Problems in RN 99

4.1 The Sobolev space H1(). . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 The Sobolev space0

H1(). . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 The Sobolev spaces Hm(), m = 2, 3, 4, . . . . . . . . . . . . . . . . . . . 104

4.4 Sobolevs inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Variational formulation of some elliptic boundaryvalue problems. . . . 107

4.5.1 (a) Homogeneous Dirichlet boundary conditions. . . . . . . . . . 107

4.5.2 (b) Homogeneous Neumann boundary conditions. . . . . . . . . 110

5 The Galerkin Finite Element Method for Elliptic BoundaryValue

Problems 112

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2 Piecewise linear, continuous functions on a triangulation of a plane

polygonal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Implementation of the finite element method with P1 triangles . . . . . 128

6 The Galerkin Finite Element Method for the Heat Equation 139

6.1 Introduction. Elliptic projection . . . . . . . . . . . . . . . . . . . . . . 139

6.2 Standard Galerkin semidiscretization . . . . . . . . . . . . . . . . . . . 141

6.3 Full discretization with the implicit Euler and the Crank-Nicolson method147

6.4 The explicit Euler method. Inverse inequalities and stiffness . . . . . . 154

7 The Galerkin Finite Element Method for the Wave Equation 161

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2 Standard Galerkin semidiscretization . . . . . . . . . . . . . . . . . . . 162

iii

7.3 Fully discrete schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

References 179

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Chapter 1

Some Elements of Hilbert Space

Theory

1.1 Vector spaces

A set V of elements u, v, w, . . . is called a vector space (over the complex numbers) if

1. For every pair of elements u V , v V we define a new element w V , their

sum, denoted by w = u+ v.

2. For every complex number and every u V we define an element w = u V ,

the product of and u.

3. Sum and product obey the following laws:

i. u, v V : u+ v = v + u.

ii. u, v, w V : (u+ v) + w = u+ (v + w).

iii. 0 V such that u+ 0 = u, u V .

iv. u V (u) V such that u+ (u) = 0.

v. 1 u = u, u V .

vi. (u) = ()u, u V and complex , .

vii. (+ )u = u+ u, for u V , , complex.

viii. (u+ v) = u+ v, for u, v V , complex.

1

The elements u, v, w, . . . of V are called vectors.

An expression of the form

1u1 + 2u2 + . . .+ nun,

where i complex numbers and ui V is called a linear combination of the vectors ui,

1 i n. The vectors u1, . . . , un are called linearly dependent if there exist complex

numbers i, not all zero, for which:

1u1 + 2u2 + . . .+ nun = 0.

They are called linearly independent if they are not linearly dependent, i.e. if 1u1 +

2u2 + . . .+ nun = 0 holds only in the case 1 = 2 = . . . = n = 0.

A vector space V is called finitedimensional (of dimension n) if V contains n

linearly independent elements and if any n+1 vectors in V are linearly dependent. As

a consequence, a set of n linearly independent vectors forms a basis of V , i.e. it is a set

of linearly independent vectors that spans V , i.e. such that any u in V can be written

uniquely as a linear combination of the basis vectors.

1.2 Inner product, Norm

A vector space V is called an inner product space if for every pair of elements u V ,

v V we can define a complex number, denoted by (u, v) and called the inner product

of u and v, with the following properties:

1. u V : (u, u) 0. If (u, u) = 0 then u = 0.

2. (u, v) = (v, u), u, v V , where z is the complex conjugate of the complex

number z.

3. (u+ v, w) = (u,w) + (v, w) for u, v, w V , , complex.

As a consequence of (2) and (3) (u, u) = (u, v) for u, v V and complex. The

vectors u, v are called orthogonal if (u, v) = 0.

For every u V we define the nonnegative number u by

u = (u, u)12 ,

2

which is called the norm of u. As a consequence of the properties of the inner product

we see that:

i. u 0 and if u = 0, then u = 0

ii. complex , u V : u = || u

iii. u, v V : u+ v u+ v (Triangle inequality).

To prove (iii) we first prove the CauchySchwarz Inequality:

iv. |(u, v)| u v, u, v V .

To prove (iv) we may assume that (u, v) = 0. We let now = (u,v)|(u,v)| . We find then for

any real that

0 (u+ v, u+ v) = 2(v, v) + 2 |(u, v)|+ (u, u).

Hence for any real the quadratic on the right hand side of the above inequality is

nonnegative. Hence, necessarily,

|(u, v)| 2 (u, u) (v, v),

which gives (iv). To prove now the triangle inequality (iii) we see that

u+ v 2 = (u+ v, u+ v) = (u, u) + (v, v) + (u, v) + (v, u)

u2 + v2 + 2 |(u, v)| u2 + v2 + 2 u v

= (u+ v) 2,

from which (iii) follows. (Supplied only with a norm that just satisfies properties

(i)(iii), V becomes a normed vector spac