Numerical Approximation of

the Ohta–Kawasaki Functional

Quentin Parsons

Kellogg College

University of Oxford

A thesis submitted for the degree of

M.Sc. in Mathematical Modelling and Scientific Computing

Trinity 2012

Acknowledgements

First and foremost, I would like to say a special word of thanks to my de-

partmental and thesis project supervisor, Prof. Endre Suli. His assistance

in crafting the results presented in Chapter 2 and his patient explanation

of various aspects of the problem to me are very much appreciated. His

enthusiasm for his subject and his vast, yet seemingly instantly available

knowledge of this branch of mathematics are consistently awe-inspiring.

I’m also indebted to him for repeatedly reading and reviewing my clumsy

attempts at type-setting this document.

I would also like to express my gratitude to Dr Kathryn Gillow for her

help, guidance and faith in me throughout this year at Oxford. Her help

in uncovering the source of ‘the missing mass’ in my finite element code

was instrumental in keeping me sane and on track at a very trying time.

This publication was based on work supported in part by Award No KUK-

C1-013-04, made by King Abdullah University of Science and Technology

(KAUST).

Abstract

We consider the Ohta–Kawasaki functional as a model for the free energy

of a diblock copolymer melt.

Following a brief overview of the physical parameters and quantities in-

volved, we derive the related Ohta–Kawasaki dynamic equation using a

suitable gradient flow method, and exhibit the result as a coupled system

of partial differential equations. Mass conservation is demonstrated after

which boundedness and stability proofs are presented in detail.

We then examine the time-sequence of finite element approximations to

the evolution problem. Using the weak form analysis as a model, we es-

tablish mass conservation, boundedness and finally, stability. We defer

convergence and error analysis of the numerical method, but we do im-

plement it in Matlab in one and two spatial dimensions. Full details

of the numerical algorithm are presented, including a simple, fixed point

method for resolving the non-linearity in the problem.

Thereafter, we use the Matlab code to study the effects of varying the

mass and non-local energy coefficient in two dimensions and find that our

results are consistent with those derived by other numerical methods, as

well as physical laboratory experiments conducted with diblock copoly-

mers.

We conclude with a set of suggestions for further work on the problem.

A website (http://people.maths.ox.ac.uk/parsons/) was created to accom-

pany this thesis. It includes links to animations of the two dimensional

evolution simulations, a full specification of the Matlab implementation

and finally, results of some of the one dimensional experiments that were

conducted.

This on-line version includes corrections made to the hardcopy that was

submitted to the Oxford Exam Schools on 6 September, 2012.

Contents

1 Introduction 1

1.1 Statement of the Ohta–Kawasaki functional . . . . . . . . . . . . . . 2

1.2 Deriving the Ohta–Kawasaki functional . . . . . . . . . . . . . . . . . 3

2 Analysis 5

2.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The dynamic equation by H−1(Ω) gradient flow . . . . . . . . . . . . 7

2.3 The weak form of the dynamic equation . . . . . . . . . . . . . . . . 8

2.3.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.3 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 The finite element approximation . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.3 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Implementation 36

3.1 A specific numerical scheme . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Results summary: two dimensional space . . . . . . . . . . . . . . . . 40

3.2.1 The effect of varying mass . . . . . . . . . . . . . . . . . . . . 40

3.2.2 The effect of varying the non-local energy coefficient . . . . . . 44

4 Conclusion 49

4.1 Opportunities for further study . . . . . . . . . . . . . . . . . . . . . 49

References 51

i

A Useful Mathematical Results 54

A.1 Gronwall’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2 Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.3 The Holder inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.4 Other useful identities and inequalities . . . . . . . . . . . . . . . . . 57

B Calculation Essentials 59

B.1 Derivation of the dynamic equation . . . . . . . . . . . . . . . . . . . 59

B.2 Approach and results highlights: the weak form . . . . . . . . . . . . 61

B.3 Approach and results highlights: the finite element approximation . . 62

C Supplementary Results and Graphs 63

C.1 Varying mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

C.1.1 Computational effort/cost . . . . . . . . . . . . . . . . . . . . 64

C.1.2 Component energy evolution . . . . . . . . . . . . . . . . . . . 65

C.2 Varying the non-local energy coefficient . . . . . . . . . . . . . . . . . 66

C.2.1 The initial condition . . . . . . . . . . . . . . . . . . . . . . . 66

C.2.2 Computational effort/cost . . . . . . . . . . . . . . . . . . . . 66

C.2.3 Total energy evolution . . . . . . . . . . . . . . . . . . . . . . 68

C.2.4 The extended run for σ = 2 . . . . . . . . . . . . . . . . . . . 68

C.2.5 h-independence . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ii

List of Figures

1.1 Diblock copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Diblock copolymer phase diagram (adapted from [mit12]) . . . . . . . 4

2.1 The orthogonal projection of the initial condition u0 ∈ H1(Ω) onto Vh 20

3.1 Typical initial conditions (2D) . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Typical ‘metastable’ end-state solutions (2D) for various m . . . . . . 42

3.3 Typical total free energy evolution over time (2D) . . . . . . . . . . . 43

3.4 SCMFT context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 End-state solutions (2D) for various σ . . . . . . . . . . . . . . . . . . 47

3.6 Competing free energy components (2D) for various σ. The red circles

depict the component energy levels of the initial condition. . . . . . . 48

B.1 Argument summary – weak form boundedness and stability . . . . . 61

B.2 Argument summary – finite element approximation boundedness and

stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C.1 The two finite element grids (2D) that were used . . . . . . . . . . . 63

C.2 Non-linear work (2D) for various m . . . . . . . . . . . . . . . . . . . 64

C.3 Typical component energy evolution over time (2D) . . . . . . . . . . 65

C.4 Initial condition used to test the effect of varying σ . . . . . . . . . . 66

C.5 Non-linear work (2D) for various σ . . . . . . . . . . . . . . . . . . . 67

C.6 Total free energy evolution (2D) for various σ . . . . . . . . . . . . . 68

C.7 Extended run results for σ = 2 (2D) . . . . . . . . . . . . . . . . . . . 69

C.8 Results for σ = 200 (2D) after 500 time steps on two different meshes 71

iii

List of Tables

3.1 Physical and control parameters – varying mass in two dimensions . . 40

3.2 Physical and control parameters – varying σ in two dimensions . . . . 44

iv

Chapter 1

Introduction

A diblock coplymer is a linear molecule comprising two subchains of what we will

term ‘type A’ and ‘type B’ monomers that repel each other, but that are joined co-

valently. This strong bond prevents them from breaking apart. In this project, we

focus on what happens as these materials are rapidly cooled. At the molecular level

according to [Cho03], ‘below a critical temperature, even a weak repulsion between

unlike monomers A and B induces a strong repulsion between the subchains caus-

ing [them] to segregate’. It is this repulsive action that gives these molecules their

remarkable ability to rearrange into ordered, periodic structures. A simple diblock

copolymer and the way in which these molecules rearrange themselves is depicted in

Fig. 1.1. A large collection of such molecules is called a ‘melt’.

Figure 1.1: Diblock copolymers

As will become apparent presently, studies of these materials have led to models

that can accurately predict the morphologies into which they will rearrange, based

on a few simple physical parameters. As such, these materials are often referred

1

to as ‘designer soft materials’ ([BF99]) with applications as diverse as upholstery

and bedding to computer memory. The book by Hamley ([Ham03]) provides an

excellent background to these materials – indeed, given their commercial application,

the literature related to polymers is vast. Other variants such as triblock and branched

copolymers (see [BF99]) exist but we restrict our attention to the diblock case here.

1.1 Statement of the Ohta–Kawasaki functional

The physical tendency of copolymer melts is to rearrange themselves in such a way as

to minimise their free energy by forming structures that minimise contacts between

the unlike monomers ([Cho03]). We model this free energy with the (scaled) Ohta–

Kawasaki functional (originally derived in [OK86]) which we now present. In doing

so, we are introduced to all of the important physical parameters:

E(u) = 12

∫Ω

(ε2 |∇u|2 + 1

2

(1− u2

)2+ σ

∣∣(−∆)−1/2(u−m)∣∣2) dΩ. (1.1)

We start by designating the type A monomer density1 in the melt by a; then the

density of type B monomers is clearly 1− a (if we assume incompressibility), and the

difference between the type A and type B monomer densities is given by 2a− 1. We

label this quantity the ‘mass’, m, of the system so that

a =m+ 1

2(1.2)

where a and m are parameters that describe the melt on average. Note that we

assume that all of the molecules in the melt have exactly the same composition.

The function u = u(x, t) in (1.1) describes the difference between the type A and

type B monomer densities as a function of space and time. It is clear that we may

describe a particular melt as comprising (on average), a certain proportion of type A

or B monomers, but obviously at small scales we will have regions where we will find

‘only type A’ or ‘only type B’ monomers. We therefore identify u(x, t) = 1 with the

(local) presence of type A monomers and u(x, t) = −1 with the presence of type B

monomers. E(u) is then a functional (of u) that describes the total free energy in the

melt.

It is obvious that the average of u over the space occupied by the melt is equivalent

to its ‘mass’ and so intuitively we expect that this mass should be conserved as the

1Equivalently, we can think of this quantity as the length of the type A monomer chain as a

proportion of the total length of each diblock copolymer macromolecule.

2

molecules rearrange themselves. These ideas are formalised later in Equations (2.2)

and (2.18). Note that (1.1) is non-dimensional and has been re-scaled in space so that

we work on the unit cube (which we denote Ω and model as (0, 1)d for d = 1, 2, 3)

whereas the melt physically occupies the volume D.

Additionally, ε represents the (scaled) interfacial thickness at the A and B monomer

intersections ([CR03]), and depends on various physical parameters and characteris-

tics of the melt according to

ε2 =l2

3a(1− a)χ|D|2/3. (1.3)

We have the following interpretations:

• |D| is the volume of the physical domain which the melt occupies;

• l (a dimensional parameter) is the so-called Kuhn statistical length measuring

the average distance between two adjacent monomers or ‘the average monomer

space size’ ([CPW09]); and

• χ is the Flory-Huggins interaction parameter, which measures the incompati-

bility of the type A and B monomers.

Finally, σ (which we will term the ‘non-local energy coefficient’) is specified by

σ =36|D|2/3

a2(1− a)2l2χN2P

. (1.4)

The new parameter here is NP , the index of polymerisation measuring the number of

monomers that occur per macromolecule.2

1.2 Deriving the Ohta–Kawasaki functional

The derivation of (1.1) was first completed by Ohta and Kawasaki ([OK86]) using

mean field theory. The Self-Consistent Mean Field Theory (‘SCMFT’) subsequently

gained prominence as the best theoretical device for predicting the behaviour of di-

block copolymers based on a and the product χNP . Matsen and Schick ([MS94]) then

developed a spectral method whereby predictions of low energy morphologies could

be made in the (a, χNP ) plane, the results of which can be seen in Fig. 1.2. A key lim-

itation of this approach, however, is that the analysis of energy minimisers follows an

2Note: we use NP instead of the usual N because the latter is used later on in our numerical

method to describe the number of time steps over which we run the finite element method.

3

ansatz-based approach, whereas (1.1) is more suited to analysing minimisers without

any such bias. Choksi and Ren showed ([CR03]) in fact that (1.1) can be regarded as

an ‘offspring’ of the SCMFT (subject to various approximations and scalings) and it

is for this reason that we use Fig. 1.2 to guide some of our work. This figure predicts

the stable physical morphology (in three dimensions) resulting from combinations of

a and χNP and exhibits an uncanny agreement with laboratory experiments. The

phases are labeled ‘L’ for lamellar, ‘G’ for gyroid, ‘C’ for hexagonally packed cylin-

ders, ‘S’ for spheres, and ‘CPS’ for close packed spheres. Structures with a < 0.5 are

formed by the Type A monomers (the Type B monomers being in the majority, fill

the free space between the ‘visible’ structures) while those resulting when a > 0.5 are

formed by the type B monomers; the latter structures are denoted with apostrophes

’ in Fig. 1.2 (as in C’, S’ and CPS’).

Figure 1.2: Diblock copolymer phase diagram (adapted from [mit12])

A two-dimensional version of Fig. 1.2 would depict lamellae for small m, spots

(approximate circles) for non-zero m and disorder for small χNP (see [CMW11]).

We proceed now to explore various theoretical aspects of the functional in (1.1)

and its finite element approximation.

4

Chapter 2

Analysis

2.1 Context

We start by defining the mathematical context within which the analysis will be

performed.1 For the time being, we largely ignore the physical significance of the

various components of the problem but focus instead on a mathematical analysis

of the underlying equations. Our starting point is the Ohta–Kawasaki Free Energy

Functional (‘OKFEF’) given above2 in (1.1) as

E(u) = 12

∫Ω

(ε2 |∇u|2 + 1

2

(1− u2

)2+ σ

∣∣(−∆N)−1/2(u−m)∣∣2) dΩ

where

−∫

Ω

u dΩ :=1

|Ω|

∫Ω

u dΩ = m (2.2)

1In this chapter, the notation ‘‖·‖’ will imply the L2(Ω)-norm and similarly, ‘(·, ·)’ with no

subscript will refer to the L2(Ω)-inner product.2An alternate form of the OKFEF is

E(u) = 12

∫Ω

(ε2 |∇u|2 + 1

2

(1− u2

)2+ σ |∇v|2

)dΩ (2.1)

where −∆v = u−m. It is possible to make sense of this definition of the OKFEF, but in order to do

so, conditions need to be specified on v to ensure the existence of a unique solution to the problem

−∆v = u−m. Specifically, if we require

∂v

∂n

∣∣∣∣∂Ω

= 0

then it is possible to specify a function space H2(Ω)∩H1∗ (Ω) (see Equation (2.4)) within which the

Lax-Milgram Theorem assures us of the existence of a unique solution to this linear problem.

5

for the unit d-cube Ω = (0, 1)d, d = 1, 2, 3, and we require

∂u

∂n

∣∣∣∣∂Ω

= 0. (2.3)

We now formally define the operator (−∆N)−1/2. In preparation for this, we consider

the space

H1∗ (Ω) =

u ∈ H1(Ω) :

∫Ω

u dΩ = 0

(2.4)

where, as usual, we use Hk(Ω) to denote the Sobolev space W k,2(Ω) of all locally

summable functions u : Ω → R with the property that Dαu ∈ L2(Ω), in the weak

sense, for |α| ≤ k (see [Eva98], Section 5.2.2).

The subscript ‘N ’ on the operator (−∆N)−1/2 specifies that it ‘carries’ with it a

zero Neumann boundary condition (without which writing down a negative power of

the Laplace operator is meaningless) such that if

(−∆N)−1w = g ⇒ −∆Ng = w with w ∈ H1∗ (Ω), (2.5)

we must have

∂g

∂n

∣∣∣∣∂Ω

= 0.

We specify that the operator (−∆N)−1/2 should act in such a way that the so-called

‘H−1(Ω) inner product’ can be written in one of the following three equivalent forms:

〈w, v〉H−1(Ω) :=

((−∆N)−1w, v)((−∆N)−1/2w, (−∆N)−1/2v

)(w, (−∆N)−1v)

∀ w, v ∈ H1∗ (Ω). (2.6)

In this context, we define H−1(Ω) as the dual of the space H1∗ (Ω); the reason for our

interest in this space is explained in Section 2.2. Specifically, if w = v, we define

〈w,w〉H−1(Ω) := ‖w‖2H−1(Ω) and then we can write

‖w‖2H−1(Ω) =

((−∆N)−1w,w

)=((−∆N)−1/2w, (−∆N)−1/2w

)=∥∥(−∆N)−1/2w

∥∥2

=

∫Ω

∣∣(−∆N)−1/2w∣∣2 dΩ (2.7)

and if u(·, t) ∈ H1(Ω) then u−m ∈ H1∗ (Ω) by virtue of (2.2). A comparison of (2.7)

and (1.1) thus reveals that we can write the OKFEF as

E(u) = 12

∫Ω

(ε2 |∇u|2 + 1

2

(1− u2

)2)

dΩ +σ

2‖u−m‖2

H−1(Ω) . (2.8)

6

Our approach in analysing the OKFEF in the forthcoming sections is summarised in

Fig. B.1 on p. 61. This figure need not be consulted at all, but could prove useful as

a map in what follows.

2.2 The dynamic equation by H−1(Ω) gradient flow

Given any initial state u(x, 0), the system is assumed to evolve in such a way that

u ∈ arg minv∈H1(Ω)E(v) := 12

∫Ω

ε2 |∇v|2 + 12

(1− v2

)2dΩ + 1

2σ ‖v −m‖2

H−1(Ω).

It is clear that we need an expression for ut to model the system evolution, and we

get the weak form of this by setting

〈ut, φ〉H−1(Ω) = −〈DE(u), φ〉 ∀ φ ∈ H1∗ (Ω), (2.9)

where DE(u) is the ‘variational’ or Gateaux derivative of the functional E(u) defined

by

〈DE(u), φ〉 = limα→0

E(u+ αφ)− E(u)

α. (2.10)

We use the H−1(Ω) inner product on the left-hand side of (2.9) because using an

alternative (for instance, the L2(Ω) inner product) nullifies the mass conservation

property we establish in Section 2.3.1 (see [Zha06]). Now,

〈DE(u), φ〉 = limα→0

1

2α

[∫Ω

(ε2 |∇ (u+ αφ)|2 + 1

2

(1− (u+ αφ)2)2

−ε2 |∇u|2 − 12

(1− u2

)2)

dΩ

+12σ ‖u+ αφ−m‖2

H−1(Ω) −12σ ‖u−m‖2

H−1(Ω)

]. (2.11)

At this point, we observe that the term ‖u+ αφ−m‖2H−1(Ω) above is meaningful since

we can write

‖u+ αφ−m‖2H−1(Ω) =

∥∥∥∥ (u−m)︸ ︷︷ ︸∈ H1

∗ (Ω)

+ αφ︸︷︷︸∈ H1

∗ (Ω)

∥∥∥∥2

H−1(Ω)

,

i.e. u+αφ−m ∈ H1∗ (Ω) as is required for our definition of the H−1(Ω) inner product

and its induced norm ‖·‖H−1(Ω) to make sense. We now define

N (u) = u(u2 − 1) (2.12)

7

and after much fairly straightforward manipulation and the application of Equation

(2.6) (see Appendix B.1 for full details) we arrive at

〈DE(u), φ〉 = −∫

Ω

φε2∆Nu−N (u)− σ(−∆N)−1

(u−m

)dΩ. (2.13)

If we now substitute (2.13) into (2.9) and rearrange, we have

〈ut, φ〉H−1(Ω) −∫

Ω

φε2∆Nu−N (u)− σ(−∆N)−1

(u−m

)dΩ = 0 ∀ φ ∈ H1

∗ (Ω).

Combining this result with the definition noted in (2.6) we obtain∫Ω

φ

(−∆N)−1ut − ε2∆Nu+N (u) + σ(−∆N)−1(u−m

)dΩ = 0 ∀ φ ∈ H1

∗ (Ω)

which means that

(−∆N)−1ut − ε2∆Nu+N (u) + σ(−∆N)−1(u−m

)= 0.

Finally, if we apply the operator (−∆N) to both sides of the previous equation, we

are led to the Ohta–Kawasaki Dynamic Equation (‘OKDE’)

ut + ∆N

(ε2∆Nu−N (u)

)+ σ(u−m) = 0. (2.14)

2.3 The weak form of the dynamic equation

We start with (2.14) and introduce a new function w = −ε2∆u + N (u) so that we

can write the OKDE as the coupled system

ut −∆w + σ(u−m) = 0

w = −ε2∆u+N (u)

in Ω× (0, T ] (2.15)

with

∂u

∂n

∣∣∣∣∂Ω

= 0 and∂w

∂n

∣∣∣∣∂Ω

= 0 on (0, T ] (2.16)

and

u(x, 0) = u0(x), x ∈ Ω, (2.17)

8

with u0 ∈ H2(Ω). Additionally, we define3

m =1

|Ω|

∫Ω

u0 dΩ

(:= −∫

Ω

u0 dΩ

)(2.18)

and note that

N (u) ≡ Φ′(u) where Φ(u) = 14(1− u2)2. (2.19)

2.3.1 Mass conservation

We start by demonstrating that the system defined above exhibits mass conservation

and in so doing, illustrate the importance and utility of the chosen boundary condition

on u and w. We have∫Ω

(u−m)t dΩ =

∫Ω

ut dΩ (since m is a constant)

=

∫Ω

∆w dΩ− σ∫

Ω

(u−m) dΩ (from (2.15))

=

∫∂Ω

∂w

∂nds︸ ︷︷ ︸

= 0

−σ∫

Ω

(u−m) dΩ (Divergence Theorem).

It follows that

d

dt

∫Ω

(u−m) dΩ + σ

∫Ω

(u−m) dΩ = 0

which we recognise as a differential equation for∫

Ω(u−m) dΩ with solution∫

Ω

(u−m)(t) dΩ =

∫Ω

(u0 −m) dΩ

︸ ︷︷ ︸

= 0 by (2.18)

e−σt, ∀ t ∈ [0, T ]. (2.20)

The implication of this result is that

m =1

|Ω|

∫Ω

u(x, t) dΩ =1

|Ω|

∫Ω

u0 dΩ ∀ t ∈ [0, T ]; (2.21)

i.e. the model (and specifically its zero Neumann spatial boundary condition) auto-

matically implies the conservation of relative concentration of monomer types over

time. This agrees with physical intuition since the process whereby diblock copoly-

mers rearrange themselves does not result in the creation or destruction of type A

or B monomers. Formally, we see that the coupled system of equations is consistent

with the definition expressed in (2.2), as required.

3We will see presently that the forthcoming definition is entirely consistent with that given in

(2.2) i.e. the system we work with exhibits ‘mass conservation’ as it should.

9

2.3.2 Formulation

We develop the two-part weak form by taking an L2(Ω)-inner product between each

of the two equations in (2.15) and a test function v ∈ H1(Ω). For the first equation

we seek u(·, t) ∈ H1(Ω) such that

(ut, v)−∫

Ω

v∆w dΩ + σ

∫Ω

(u−m)v dΩ = 0 ∀ v ∈ H1(Ω)

or

(ut, v) + (∇w,∇v) + σ (u−m, v) = 0 ∀ v ∈ H1(Ω) and t ∈ (0, T ], (2.22)

after using Theorem 9, p. 58. For the second equation in (2.15), we note the zero

Neumann boundary condition on u, and recalling that N (u) = Φ′(u), obtain

(w, v) = ε2 (∇u,∇v) + (Φ′(u), v) ∀ v ∈ H1(Ω).

So in this case, we seek w(·, t) ∈ H1(Ω) such that

(w, v) = ε2 (∇u,∇v) + (Φ′(u), v) ∀ v ∈ H1(Ω) and t ∈ (0, T ]. (2.23)

2.3.3 Boundedness

We will now show that solutions u(·, t) to Equations (2.22) and (2.23) are bounded

in the L∞(Ω) and H2(Ω) norms for all t ∈ [0, T ]. To start, we take v = w in (2.22)

to see that

(ut, w) + ‖∇w‖2 + σ

∫Ω

(u−m)w dΩ = 0. (2.24)

Then, setting v = ut in (2.23) we have

(w, ut) = ε2 (∇u,∇ (ut)) + (Φ′(u), ut)

⇒ (ut, w) = 12ε2

d

dt‖∇u‖2 +

d

dt(Φ(u), 1) . (2.25)

Subtracting (2.24) from (2.25) and rearranging then gives

12ε2

d

dt‖∇u‖2 +

d

dt(Φ(u), 1) + ‖∇w‖2 + σ

∫Ω

(u−m)w dΩ = 0. (2.26)

We consider the last term in (2.26):∫Ω

(u−m)w dΩ =

(u−−

∫Ω

u dΩ, w −−∫

Ω

w dΩ

)(using (2.20) and (2.21))

≤∥∥∥∥u−−∫

Ω

u dΩ

∥∥∥∥∥∥∥∥w −−∫Ω

w dΩ

∥∥∥∥ (Cauchy–Schwarz ineq.)

≤ c2P ‖∇u‖ ‖∇w‖ (2.27)

10

using Poincare’s inequality (Theorem 8, p. 57). Using (2.27) in (2.26), we have

12ε2

d

dt‖∇u‖2 +

d

dt(Φ(u), 1) + ‖∇w‖2 ≤ σc2

P ‖∇u‖ ‖∇w‖

≤ 12σ2c4

P ‖∇u‖2 + 1

2‖∇w‖2

by Young’s inequality (Theorem 7, p. 57), and so

ε2d

dt‖∇u‖2 + 2

d

dt(Φ(u), 1) + ‖∇w‖2 ≤ σ2c4

P ‖∇u‖2 .

Integrating this in time over [0, t] and rearranging gives

ε2 ‖∇u(t)‖2 + 2 (Φ(u(t)), 1) +

∫ t

0

‖∇w(s)‖2 ds ≤ ε2 ‖∇u0‖2 + 2 (Φ(u0), 1)

+σ2c4

P

ε2

∫ t

0

ε2 ‖∇u(s)‖2 + 2 (Φ(u(s)), 1) +

∫ s

0

‖∇w(r)‖2 dr

ds,

where we have added non-negative terms on the right-hand side in preparation for

Gronwall’s Lemma (Theorem 1, p. 54). Using this result, we see that

ε2 ‖∇u(t)‖2 + 2 (Φ(u(t)), 1) +

∫ t

0

‖∇w(s)‖2 ds

≤ε2 ‖∇u0‖2 + 2 (Φ(u0), 1)

exp

(σ2c4

P

ε2t

)≤ C (ε, σ, u0, cP , T ) ∀ t ∈ [0, T ]. (2.28)

Consequently, ∫ t

0

‖∇w(s)‖2 ds ≤ C (ε, σ, u0, cP , T ) (2.29)

and also

maxt∈[0,T ]

‖∇u(t)‖ ≤ C (ε, σ, u0, cP , T ) . (2.30)

We now take v = ut in (2.22) and arrive at

‖ut‖2 + (∇w,∇ut) + σ

∫Ω

(u−m)ut dΩ = 0. (2.31)

If we differentiate (2.23) with respect to t, we obtain

(wt, v) = ε2 (∇ut,∇v) + (Φ′′(u)ut, v)

11

and so setting v = w in this latter equation yields

12

d

dt‖w‖2 = ε2 (∇w,∇ut) + (Φ′′(u)ut, w) . (2.32)

Adding (2.32) to ε2 times (2.31) then gives

ε2 ‖ut‖2 + 12

d

dt‖w‖2 + 1

2σε2

d

dt

∫Ω

(u−m)2 dΩ = (Φ′′(u)ut, w)

or

12

d

dt

[‖w‖2 + σε2

∫Ω

(u−m)2 dΩ

]+ ε2 ‖ut‖2 =

((3u2 − 1)ut, w

)(2.33)

since Φ′′(u) = 3u2 − 1. Now((3u2 − 1)ut, w

)≤ 3

∥∥u2w∥∥ ‖ut‖+ ‖w‖ ‖ut‖ (Cauchy–Schwarz ineq.)

≤(

9

ε2∥∥u2w

∥∥2+ε2

4‖ut‖2

)+

(1

ε2‖w‖2 +

ε2

4‖ut‖2

)=ε2

2‖ut‖2 +

9

ε2∥∥u2w

∥∥2+

1

ε2‖w‖2 ,

where we used Young’s inequality (Theorem 7, p. 57) twice to obtain the second line.

If we substitute this into the right-hand side of (2.33), rearrange and make use of

Holder’s inequality (Theorem 5, p. 56) with conjugate exponents p = 3/2 and q = 3,

we see that

d

dt

[‖w‖2 + σε2 ‖u−m‖2]+ ε2 ‖ut‖2 ≤ 18

ε2‖u‖4

L6(Ω) ‖w‖2L6(Ω) +

2

ε2‖w‖2

≤ 18

ε2c6S ‖u‖

4H1(Ω) ‖w‖

2H1(Ω) +

2

ε2‖w‖2 ,

where we used Sobolev’s inequality (Theorem 3, p. 55) in d = 1, 2, 3 dimensions to

obtain the last line. Integrating this in time over [0, t] and noting the definition of

the H1(Ω)-norm in the ‖w‖2H1(Ω) term, leads to

‖w(t)‖2 + σε2 ‖u(t)−m‖2 + ε2∫ t

0

‖ut‖2 ds

≤ ‖w0‖2 + σε2 ‖u0 −m‖2 +18c6

S

ε2maxt∈[0,T ]

‖u‖4H1(Ω)

∫ t

0

‖∇w(s)‖2 ds

+

(2

ε2+

18c6S

ε2maxt∈[0,T ]

‖u‖4H1(Ω)

)∫ t

0

‖w(s)‖2 + σε2 ‖u(s)−m‖2 + ε2

∫ s

0

‖ut‖2 dr

ds,

where, as in the lead-up to Equation (2.28), we have again added non-negative terms

on the right-hand side in preparation for Gronwall’s Lemma (Theorem 1, p. 54).

12

Using this result again, we see that, for all t ∈ (0, T ],

‖w(t)‖2 + σε2 ‖u(t)−m‖2 + ε2∫ t

0

‖ut‖2 ds

≤(‖w0‖2 + σε2 ‖u0 −m‖2 +

18c6S

ε2maxt∈[0,T ]

‖u‖4H1(Ω)

∫ t

0

‖∇w(s)‖2 ds

)× exp

(2

ε2+

18c6S

ε2maxt∈[0,T ]

‖u‖4H1(Ω)

)t

. (2.34)

Now

‖u‖2H1(Ω) = ‖u−m+m‖2 + ‖∇u‖2

≤ 2

∥∥∥∥u−−∫Ω

u dΩ

∥∥∥∥2

+ 2

∫Ω

m2dΩ + ‖∇u‖2 (Equation A.6, p. 57)

≤(2c2P + 1

)‖∇u‖2 + 2m2

∫Ω

dΩ (Poincare’s ineq. – Thm. 8, p. 57)

=(2c2P + 1

)‖∇u‖2 + 2m2|Ω|

≤ C (ε, σ, u0, cP , T,m, |Ω|) ,

where we used (2.30) in the last step to eliminate ‖∇u‖2. From this result and (2.29),

we see that all of the terms on the right-hand side of (2.34) are bounded by a constant;

specifically

‖w(t)‖2 + σε2 ‖u(t)−m‖2 + ε2∫ t

0

‖ut‖2 ds

≤ C (w0, ε, σ, u0, cP , cS, T,m, |Ω|) . (2.35)

Since w0 := −ε2∆u0 +N (u0) depends on ε and u0 only, we can suppress w0 in (2.35)

and since all of the terms on the left-hand side of (2.35) are non-negative, we see that

‖w(·, t)‖ ≤ C (ε, σ, u0, cP , cS, T,m, |Ω|) (2.36)

and also

‖u(·, t)−m‖ ≤ C (ε, σ, u0, cP , cS, T,m, |Ω|) . (2.37)

Now as in [S12], we set v = ∆u and then,

(∇u,∇(∆u)) = (∇u,∇v)

= (−∆u, v) (Theorem 9, p. 58)

= −‖∆u‖2 (2.38)

13

since ∂u/∂n|∂Ω = 0. As a result, setting v = ∆u in (2.23) gives

(w,∆u) = −ε2 ‖∆u‖2 + (Φ′(u),∆u) .

Consequently,

ε2 ‖∆u‖2 ≤ (‖w‖+ ‖Φ′(u)‖) ‖∆u‖ ,

which implies that

ε2 ‖∆u‖ ≤ ‖w‖+∥∥u3 − u

∥∥≤ ‖w‖+ ‖u‖+ ‖u‖3

L6(Ω)

≤ ‖w‖+ ‖u‖+ c3S ‖u‖

3H1(Ω)

using Sobolev’s inequality (Theorem 3, p. 55). So we have

‖∆u‖ ≤ 1

ε2

(‖w‖+ ‖u‖+ c3

S ‖u‖3H1(Ω)

)for any t ∈ [0, T ] and thus

maxt∈[0,T ]

‖∆u‖ ≤ 1

ε2

(maxt∈[0,T ]

‖w‖+ maxt∈[0,T ]

‖u−m‖+ ‖m‖+ c3S maxt∈[0,T ]

(‖u‖2 + ‖∇u‖2)3

)≤ 1

ε2

(maxt∈[0,T ]

‖w‖+ maxt∈[0,T ]

‖u−m‖+m |Ω|1/2

+c3S maxt∈[0,T ]

(2 ‖u−m‖2 + 2m2|Ω|+ ‖∇u‖2

)3)

≤ C (ε, σ, u0,m, cS, cP , T, |Ω|) ,

where we use Equation A.6, p. 57 and then apply (2.36) to ‖w‖, (2.37) to ‖u−m‖and (2.30) to ‖∇u‖2.

Assuming that ∂Ω is of class C2 or that Ω is a convex polygonal (for d = 2) or poly-

hedral (for d = 3) domain, we have by Theorem 10, p. 58 that maxt∈[0,T ] ‖u(·, t)‖H2(Ω) ≤C maxt∈[0,T ] ‖∆u‖ for some C > 0, and so

maxt∈[0,T ]

‖u(·, t)‖H2(Ω) ≤ C (ε, σ, u0,m, cS, cP , T, |Ω|) . (2.39)

By Sobolev embedding, which says that the H2(Ω)-norm bounds the infinity-norm

for d = 1, 2, 3 (Theorem 4, p. 55), we therefore have

maxt∈[0,T ]

‖u(·, t)‖L∞(Ω) ≤ C (ε, σ, u0,m, cS, cP , T, |Ω|) . (2.40)

14

2.3.4 Stability

Now that we have established the boundedness of the (weak form of the) PDE, we

move on to show its stability. For convenience we adopt the following simple notation:

we suppose that the weak solution can be expressed as a pair of functions U = (u,w)

that solve (2.22) and (2.23).

Suppose then that the problem has two solutions U1 = (u1, w1) and U2 = (u2, w2).4

For the solution U1 we have

(u1,t, v) + (∇w1,∇v) + σ (u1 −m, v) = 0 ∀ v ∈ H1(Ω)

and

(w1, v) = ε2 (∇u1,∇v) + (Φ′(u1), v) ∀ v ∈ H1(Ω).

Similarly for U2 we can write

(u2,t, v) + (∇w2,∇v) + σ (u2 −m, v) = 0 ∀ v ∈ H1(Ω)

and

(w2, v) = ε2 (∇u2,∇v) + (Φ′(u2), v) ∀ v ∈ H1(Ω).

Subtracting these two problems leads to the following problem in u ≡ u1 − u2 and

w ≡ w1 − w2 for which we define U = (u,w) = (u1 − u2, w1 − w2) = U1 − U2:

find u(·, t) ∈ H1(Ω) s.t. (ut, v) + (∇w,∇v) + σ (u, v) = 0 ∀ v ∈ H1(Ω) (2.41)

and

find w(·, t) ∈ H1(Ω) s.t. (w, v) = ε2 (∇u,∇v) + (Φ′(u1)− Φ′(u2), v) ∀ v ∈ H1(Ω)

(2.42)

for every t ∈ (0, T ]. We notice immediately the non-linearity in the second equation

of the problem statement. To analyse this problem, we set v = u in (2.41) to obtain

ε2 12

d

dt‖u‖2 + ε2 (∇u,∇w) + ε2σ ‖u‖2 = 0 (2.43)

after multiplying through by ε2 and observing that (ut, u) = 12

∫Ω

(u2)t dΩ = 12

ddt‖u‖2.

Then, putting v = w in (2.42) gives

‖w‖2 = ε2 (∇u,∇w) + (Φ′(u1)− Φ′(u2), w) . (2.44)

4In what follows, the notation u1,t will denote the time derivative of the solution u1; the subscript

‘1’ will not imply an x (or first spatial) derivative.

15

Subtracting (2.44) from (2.43) and rearranging leads to

ε2 12

d

dt‖u‖2 + ‖w‖2 + ε2σ ‖u‖2 = (Φ′(u1)− Φ′(u2), w) . (2.45)

Now Φ′(u) = u(u2− 1) is a differentiable function and so by the mean value theorem,

Φ′(u1)− Φ′(u2) = Φ′′ (θu1 + (1− θ)u2)u

for some θ ∈ [0, 1]. Substituting this into (2.45), and noting that Φ′′(u) = 3u2 − 1

gives

ε2 12

d

dt‖u‖2 + ‖w‖2 + ε2σ ‖u‖2 =

∫Ω

uw[3 (θu1 + (1− θ)u2)2 − 1

]dΩ (2.46)

for some θ ∈ [0, 1]. According to (2.40), u1 and u2 are bounded in the infinity norm

as they are two solutions of the problem. Consequently, if we define two constants

K1 = K1(ε, σ, (u1)0,m, cS, cP , T, |Ω|) and K2 = K2(ε, σ, (u2)0,m, cS, cP , T, |Ω|) such

that

‖u1‖L∞(Ω) = K1 <∞ and ‖u2‖L∞(Ω) = K2 <∞,

then from Equation (2.46) we obtain

12ε2

d

dt‖u‖2 + ‖w‖2 + ε2σ ‖u‖2 ≤

∫Ω

|u||w|[3 (θK1 + (1− θ)K2)2 + 1

]dΩ

≤ C ‖u‖ ‖w‖ (Cauchy–Schwarz ineq.)

≤ 12‖w‖2 + 1

2C2 ‖u‖2

by Young’s inequality (Theorem 7, p. 57), and where we defined a new constant

C = 3 (θK1 + (1− θ)K2)2 + 1 = C(ε, σ, (u1)0, (u2)0,m, cS, cP , T, |Ω|).

Moving the 12‖w‖2 term over to the left, we have

12ε2

d

dt‖u‖2 + 1

2‖w‖2 + ε2σ ‖u‖2 ≤ 1

2C2 ‖u‖2 . (2.47)

At this stage, there are two cases to consider:

1) ε2σ ≥ 12C2: in this case, we can rearrange (2.47) to obtain

ε2d

dt‖u‖2 + ‖w‖2 ≤ 0

16

which we integrate in time over [0, t] to see that

ε2 ‖u(t)‖2 +

∫ t

0

‖w(s)‖2 ds ≤ ε2 ‖u0‖2

for any t ∈ [0, T ], where u0 = u(x, 0) ≡ (u1)0 − (u2)0. We conclude that

‖u(·, t)‖ ≤ ‖u0‖ for any t ∈ [0, T ],

i.e. u ∈ L∞ (0, T ;L2(Ω)) where

‖u‖L∞(0,T ;L2(Ω)) := ess sup0≤t≤T ‖u(·, t)‖ ≤ ‖u0‖ <∞. (2.48)

In addition, we have(∫ t

0

‖w(·, s)‖2 ds

)1/2

≤ ε ‖u0‖ for any t ∈ [0, T ],

and so, if we let the upper limit of integration t = T , then w ∈ L2 (0, T ;L2(Ω))

where

‖w‖L2(0,T ;L2(Ω)) :=

(∫ T

0

‖w(·, s)‖2 ds

)1/2

≤ ε ‖u0‖ <∞. (2.49)

If we now define the norm of U = (u,w) ∈ L∞ (0, T ;L2(Ω)) × L2 (0, T ;L2(Ω)) in

such a way that

‖U‖2L∞(0,T ;L2(Ω))×L2(0,T ;L2(Ω)) := ‖u‖2

L∞(0,T ;L2(Ω)) + ‖w‖2L2(0,T ;L2(Ω)) , (2.50)

then adding the squares of (2.48) and (2.49) leads to

‖U‖L∞(0,T ;L2(Ω))×L2(0,T ;L2(Ω)) ≤√

1 + ε2 ‖(u1)0 − (u2)0‖ , (2.51)

which implies that the weak form of the problem is stable if ε2σ ≥ 12C2.

2) ε2σ < 12C2: in this case, we can rearrange (2.47) to obtain

ε2d

dt‖u‖2 + ‖w‖2 ≤

(C2 − 2ε2σ

)‖u‖2 .

We integrate this in time over [0, t] and obtain

ε2 ‖u(t)‖2 +

∫ t

0

‖w(s)‖2 ds ≤ ε2 ‖u0‖2

+C2 − 2ε2σ

ε2

∫ t

0

ε2 ‖u(s)‖2 +

∫ s

0

‖w(r)‖2 dr

ds,

17

where we have added a strategic non-negative term on the right-hand side in the

usual way in preparation for the application of Gronwall’s Lemma (Theorem 1,

p. 54). In turn, this result implies that

ε2 ‖u(t)‖2 +

∫ t

0

‖w(s)‖2 ds ≤ ε2 ‖u0‖2 exp

(C2 − 2ε2σ

ε2t

)= ε2C2 ‖u0‖2 ,

where we defined another new constant

C =

√√√√exp

(C2 − 2ε2σ

ε2t

)= C(ε, σ, (u1)0, (u2)0,m, cS, cP , T, |Ω|).

As in the first case of the stability proof, we conclude that

‖u(·, t)‖ ≤ C ‖u0‖ for any t ∈ [0, T ]

i.e. u ∈ L∞ (0, T ;L2(Ω)) using the definition in (2.48). Furthermore,(∫ t

0

‖w(·, s)‖2 ds

)1/2

≤ εC ‖u0‖ for any t ∈ [0, T ],

and so, once again, w ∈ L2 (0, T ;L2(Ω)) using the definition in (2.49). Using

the definition in (2.50), we see again that U = (u,w) ∈ L∞ (0, T ;L2(Ω)) ×L2 (0, T ;L2(Ω)) and

‖U‖L∞(0,T ;L2(Ω))×L2(0,T ;L2(Ω)) ≤ C√

1 + ε2 ‖(u1)0 − (u2)0‖ , (2.52)

that is, the weak form is stable if ε2σ < 12C2.

We conclude that the weak form of the problem is stable under all conditions.

Next, we shall define a finite element approximation of the problem in (2.22) and

(2.23) with implicit Euler time-stepping. We shall then show that bounds analogous

to those derived for the weak form of the problem also hold for the numerical method.

2.4 The finite element approximation

Our approach in analysing the finite element approximation to the OKDE is sum-

marised in Fig. B.2, p. 62. It is clear that this figure, and the structure of the

underlying arguments, is very similar to that presented in Fig. B.1. As with the

analysis of the weak form, this graphical argument summary can be ignored entirely,

but could prove useful as a guide through the lengthy arguments that follow.

18

2.4.1 Formulation

Let Th be a triangulation of Ω (i.e. the spatial portion of the space-time domain)

consisting of shape-regular triangles and let Vh denote a finite-element subspace of

H1(Ω) defined on Th. Let ∆t = T/N,N ≥ 1. The finite element problem has the

following ingredients:

• A finite element approximation to the problem in physical space where unh rep-

resents the approximation of u(·, n∆t) and wnh represents the approximation of

w(·, n∆t) in Vh; and

• A backward Euler approximation of the time derivative ut.

Such a finite element approximation to the weak form of the problem can be stated

in two parts as respective counterparts to (2.22) and (2.23) as follows: we wish to

find unh, wnh ∈ Vh such that(unh − un−1

h

∆t, vh

)+ (∇wnh ,∇vh) + σ (unh −m, vh) = 0 ∀ vh ∈ Vh (2.53)

and

(wnh , vh) = ε2 (∇unh,∇vh) + (Φ′(unh), vh) ∀ vh ∈ Vh (2.54)

for n = 1, 2, . . . N where Φ′(u) = u(u2 − 1). For the sake of completeness, we remark

that the zero Neumann boundary condition specified earlier for u and w has no explicit

weak (or finite element) formulation per se. It is implicitly present in Equations (2.53)

and (2.54) in that it was used when Theorem 9 (p. 58) was applied to create the weak

forms of the equations in Equations (2.22) and (2.23).

For the initial condition, we consider what we will call the ‘H2(Ω) orthogonal

projection’ of the initial condition u0 ∈ H1(Ω) onto the space Vh, a geometric inter-

pretation of which is depicted in Fig. 2.1.

Conceptually, we require u0 − u0h to be ‘perpendicular’ to the space Vh so that

our approximation u0h of u0 in Vh is as close as possible (in some norm) to u0 itself.

In order to define ‘perpendicular’ functions, we require an inner product and to this

end, it is convenient to define the ‘H2(Ω) inner product’ on H1(Ω) as

〈u, v〉H2(Ω) = (u, v) + (∇u,∇v) + (∆hu,∆hv) for u, v ∈ H1(Ω), (2.55)

where, for u ∈ H1(Ω), we define ∆hu ∈ Vh such that

(−∆hu, vh) = (∇u,∇vh) ∀ vh ∈ Vh.

19

Figure 2.1: The orthogonal projection of the initial condition u0 ∈ H1(Ω) onto Vh

The significance of the ‘h’-subscript stems from the fact that ∆hu depends on the

discretisation being considered. We note that for a given function u ∈ H1(Ω), ∆hu ∈Vh is unique.5

Using this inner product, we define the representation u0h ∈ Vh of the initial con-

dition u0 ∈ H1(Ω) such that the following orthogonality relation holds (see Fig. 2.1):⟨u0 − u0

h, vh⟩H2

(Ω)= 0 ∀ vh ∈ Vh. (2.56)

The implication of this definition is that u0h ∈ Vh is the closest element to u0 ∈ H1(Ω)

as measured by the (induced) H2(Ω)-norm which is given by

‖u‖2H2

(Ω) = ‖u‖2 + ‖∇u‖2 + ‖∆hu‖2 . (2.57)

Equivalently, we can state the condition in Equation (2.56) as⟨u0h, vh

⟩H2

(Ω)= 〈u0, vh〉H2

(Ω) ∀ vh ∈ Vh

or, if u0 ∈ H2(Ω) with ∂u0/∂n = 0|∂Ω, as we shall henceforth suppose,(u0h, vh

)+(∇u0

h,∇vh)

+(∆hu

0h,∆hvh

)= (u0, vh) + (∇u0,∇vh) + (∆Nu0,∆hvh) ∀ vh ∈ Vh. (2.58)

5As in [S12], suppose we have some given u ∈ H1(Ω) and seek wh ∈ Vh such that

(wh, vh) = (∇u,∇vh) ∀ vh ∈ Vh.

We will show that there exists one such wh ∈ Vh and label it ‘−∆hu’. Define the linear functional

l (vh) := (∇u,∇vh) and observe by the Cauchy–Schwarz inequality that

|l (vh)| ≤ ‖∇u‖ ‖∇vh‖ ≤ ‖∇u‖ ‖vh‖H1(Ω) ≤ C(h) ‖∇u‖ ‖vh‖

since all norms on the finite-dimensional space Vh are equivalent. It follows that l is a bounded

linear functional on Vh; consequently by the Riesz Representation Theorem there exists a unique

wh ∈ Vh such that l(vh) = (wh, vh) for every vh ∈ Vh, which we label −∆hu.

20

Notice that in the last inner product on the right-hand side of (2.58) we have retrieved

(∆hu0,∆hvh) = (∆Nu0,∆hvh) because for such a function u0,

(−∆hu0, vh) = (∇u0,∇vh) = (−∆Nu0, vh) ∀ vh ∈ Vh,

by Theorem 9, p. 58 and noting that ∆hu0 ∈ Vh.

2.4.2 Mass conservation

We start by demonstrating that the sequence of finite element approximations exhibits

mass conservation. We take vh ≡ 1 in (2.53) to obtain(unh − un−1

h

∆t, 1

)+ σ (unh −m, 1) = 0.

If we now write unh − un−1h = (unh −m)− (un−1

h −m), we retrieve((unh −m)(1 + σ∆t)− (un−1

h −m), 1)

= 0

which is the same as

(1 + σ∆t) (unh −m, 1) =(un−1h −m, 1

)which, in turn, implies by induction that

(unh −m, 1) = (1 + σ∆t)−1(un−1h −m, 1

)= · · · = (1 + σ∆t)−n

(u0h −m, 1

).

However, if we set vh ≡ 1 ∈ Vh in (2.58) and note that ∆h1 = 0, then we are left with(u0h, 1)

= (u0, 1) ⇒(u0h −m, 1

)= (u0 −m, 1)

if we subtract (m, 1) from each side. We know that (u0 −m, 1) = 0 because this is

just a rearranged form of (2.18), and so

(unh −m, 1) = 0 for n = 0, 1, 2, . . . , N, (2.59)

which expresses conservation of mass in the sequence of finite element approximations.

2.4.3 Boundedness

We now show that the sequence of finite element approximations is bounded, uni-

formly in h. Taking vh = wnh in (2.53) gives(unh − un−1

h

∆t, wnh

)+ ‖∇wnh‖

2 + σ (unh −m,wnh) = 0 (2.60)

21

and then setting vh =(unh − un−1

h

)/∆t in (2.54) yields(

wnh ,unh − un−1

h

∆t

)= ε2

(∇unh,∇

unh − un−1h

∆t

)+

(Φ′(unh),

unh − un−1h

∆t

). (2.61)

Subtracting (2.60) from (2.61) and noting the symmetry of terms in the inner product

gives

ε2(∇unh,

∇unh −∇un−1h

∆t

)+

(Φ′(unh),

unh − un−1h

∆t

)+ ‖∇wnh‖

2 + σ (unh −m,wnh) = 0.

Now we re-write ∇unh in the left-most inner product of this last equation as

∇unh =∆t

2

∇unh −∇un−1h

∆t+∇unh +∇un−1

h

2

so that

ε2

2∆t

(∇(unh − un−1

h

),∇(unh − un−1

h

))+(∇unh +∇un−1

h ,∇unh −∇un−1h

)+

(Φ′(unh),

unh − un−1h

∆t

)+ ‖∇wnh‖

2 + σ (unh −m,wnh) = 0. (2.62)

If we multiply Equation (2.59) by −∫

Ωwnh dΩ ∈ R, we see that(

unh −m,−∫

Ω

wnh dΩ

)= 0 ⇒ (unh −m,wnh) =

(unh −m,wnh −−

∫Ω

wnh dΩ

)(2.63)

and so if we substitute for (unh −m,wnh) in (2.62) using (2.63) and rearrange, we

obtain

ε2

2∆t

(‖∇unh‖

2 −∥∥∇un−1

h

∥∥2)

+ε2

2∆t

∥∥∇(unh − un−1h )

∥∥2+

1

∆t

(Φ′(unh), unh − un−1

h

)+ ‖∇wnh‖

2 + σ

(unh −m,wnh −−

∫Ω

wnh dΩ

)= 0. (2.64)

From the Taylor series expansion of Φ(b) about a we determine that

Φ′(a)(a− b) = Φ(a)− Φ(b) + 12Φ′′(η)(b− a)2

for some η ∈ (a, b). We have Φ(u) = 14(1 − u2)2 so Φ′(u) = u3 − u and finally

Φ′′(u) = 3u2 − 1 ≥ −1. Hence

Φ(a)− Φ(b)− 12(b− a)2 ≤ Φ′(a)(a− b)

so certainly,

(Φ(a), 1)− (Φ(b), 1)− 12‖a− b‖2 ≤ (Φ′(a), a− b) . (2.65)

22

Consequently, if we make the associations a = unh and b = un−1h in (2.65), we obtain

(Φ(unh), 1)−(Φ(un−1

h ), 1)− 1

2

∥∥unh − un−1h

∥∥2 ≤(Φ′(unh), unh − un−1

h

).

If we substitute this for(Φ′(unh), unh − un−1

h

)in (2.64), multiply through by 2∆t, and

rearrange, we obtain

ε2 ‖∇unh‖2 + 2 (Φ(unh), 1) + ε2

∥∥∇(unh − un−1h )

∥∥2+ 2∆t ‖∇wnh‖

2

≤ ε2∥∥∇un−1

h

∥∥2+ 2

(Φ(un−1

h ), 1)

+∥∥unh − un−1

h

∥∥2 − 2σ∆t

(unh −m,wnh −−

∫Ω

wnh dΩ

)≤ ε2

∥∥∇un−1h

∥∥2+ 2

(Φ(un−1

h ), 1)

+∥∥unh − un−1

h

∥∥2

+2σ∆t ‖unh −m‖∥∥∥∥wnh −−∫

Ω

wnh dΩ

∥∥∥∥ (Cauchy–Schwarz ineq.)

≤ ε2∥∥∇un−1

h

∥∥2+ 2

(Φ(un−1

h ), 1)

+∥∥unh − un−1

h

∥∥2+ 2σ∆tc2

P ‖∇unh‖ ‖∇wnh‖

since

‖unh −m‖ =

∥∥∥∥unh −−∫Ω

unh dΩ

∥∥∥∥ ≤ cP ‖∇unh‖ and

∥∥∥∥wnh −−∫Ω

wnh dΩ

∥∥∥∥ ≤ cP ‖∇wnh‖

by Poincare’s inequality (Theorem 8, p. 57). We now apply Young’s inequality (The-

orem 7, p. 57) to the last term on the right-hand side above and subtract ∆t ‖∇wnh‖2

from both sides of the result to conclude that

ε2 ‖∇unh‖2 + 2 (Φ(unh), 1) + ε2

∥∥∇(unh − un−1h )

∥∥2+ ∆t ‖∇wnh‖

2

≤ ε2∥∥∇un−1

h

∥∥2+ 2

(Φ(un−1

h ), 1)

+∥∥unh − un−1

h

∥∥2+ σ2c4

P∆t ‖∇unh‖2 . (2.66)

Setting vh = unh − un−1h in (2.53) and multiplying through by ∆t gives∥∥unh − un−1

h

∥∥2+ ∆t

(∇wnh ,∇

(unh − un−1

h

))+ σ∆t

(unh −m,unh − un−1

h

)= 0 (2.67)

which implies by the Cauchy–Schwarz inequality that∥∥unh − un−1h

∥∥2 ≤ ∆t ‖∇wnh‖∥∥∇ (unh − un−1

h

)∥∥+ σ∆t ‖unh −m‖∥∥unh − un−1

h

∥∥≤ ∆t ‖∇wnh‖

∥∥∇ (unh − un−1h

)∥∥+ σ∆tcP ‖∇unh‖∥∥unh − un−1

h

∥∥≤ ∆t ‖∇wnh‖

∥∥∇ (unh − un−1h

)∥∥+σ2∆t2c2

P

2‖∇unh‖

2 + 12

∥∥unh − un−1h

∥∥2

using the Young and Poincare inequalities (Theorems 7, 8, p. 57). It follows that∥∥unh − un−1h

∥∥2 ≤ ∆t ‖∇wnh‖ 2∥∥∇ (unh − un−1

h

)∥∥+ σ2c2P∆t2 ‖∇unh‖

2

≤ 12∆t ‖∇wnh‖

2 + 2∆t∥∥∇ (unh − un−1

h

)∥∥2+ σ2c2

P∆t2 ‖∇unh‖2 , (2.68)

23

again, by Young’s inequality (Theorem 7, p. 57). Substituting (2.68) into (2.66) for∥∥unh − un−1h

∥∥2gives for each n that

ε2 ‖∇unh‖2 + 2 (Φ(unh), 1) + (ε2 − 2∆t)

∥∥∇(unh − un−1h )

∥∥2+ 1

2∆t ‖∇wnh‖

2

≤ ε2∥∥∇un−1

h

∥∥2+ 2

(Φ(un−1

h ), 1)

+ σ2c2P (c2

P + ∆t)∆t ‖∇unh‖2 .

Summing this through n = 1, 2, . . . , k, (k ≤ N) gives

ε2k∑

n=1

‖∇unh‖2 +2

k∑n=1

(Φ(unh), 1)+(ε2−2∆t)k∑

n=1

∥∥∇(unh − un−1h )

∥∥2+ 1

2∆t

k∑n=1

‖∇wnh‖2

≤ ε2k∑

n=1

∥∥∇un−1h

∥∥2+ 2

k∑n=1

(Φ(un−1

h ), 1)

+ σ2c2P (c2

P + ∆t)∆tk∑

n=1

‖∇unh‖2.

Canceling like terms on each side then leads to

(ε2 − σ2c2

P (c2P + ∆t)∆t

) ∥∥∇ukh∥∥2+ 2

(Φ(ukh), 1

)+ (ε2 − 2∆t)

k∑n=1

∥∥∇(unh − un−1h )

∥∥2

+ 12

k∑n=1

∆t ‖∇wnh‖2 ≤ ε2

∥∥∇u0h

∥∥2+ 2

(Φ(u0

h), 1)

+ σ2c2P (c2

P + ∆t)k−1∑n=1

∆t ‖∇unh‖2.

Note that in the above, we moved the last term in the sum on the end of the right-

hand side (i.e. σ2c2P (c2

P +∆t)∆t∥∥∇ukh∥∥2

) over to the left-hand side. If we now assume

that

∆t ≤ ε2

4⇒ ε2

2≤ ε2 − 2∆t (2.69)

and

ε2

2≤ ε2 − σ2c2

P (c2P + ∆t)∆t ⇒ σ2c2

P (c2P + ∆t)∆t ≤ ε2

2, (2.70)

then we can replace the coefficients of∥∥∇ukh∥∥2

and∑k

n=1

∥∥∇(unh − un−1h )

∥∥2on the

left-hand side with ε2/2 to obtain

ε2

2

∥∥∇ukh∥∥2+ 2

(Φ(ukh), 1

)+ε2

2

k∑n=1

∥∥∇(unh − un−1h )

∥∥2+ 1

2

k∑n=1

∆t ‖∇wnh‖2

≤ ε2∥∥∇u0

h

∥∥2+ 2

(Φ(u0

h), 1)

+2

ε2σ2c2

P (c2P + ∆t)

k−1∑n=1

∆t

(ε2

2‖∇unh‖

2

). (2.71)

24

The discrete Gronwall Lemma (Theorem 2, p. 55) then gives

12ε2∥∥∇ukh∥∥2

+ 2(Φ(ukh), 1

)+ 1

2ε2

k∑n=1

∥∥∇(unh − un−1h )

∥∥2+ 1

2

k∑n=1

∆t ‖∇wnh‖2

≤(ε2∥∥∇u0

h

∥∥2+ 2

(Φ(u0

h), 1))

exp

(2σ2c2

P (c2P + ∆t)

ε2T

)= C (ε, σ, u0, cP , T ) (2.72)

for every k = 1, 2, . . . , N . Because all of the terms on the left-hand side of (2.72)

above are non-negative, we have

k∑n=1

∆t ‖∇wnh‖2 ≤ C (ε, σ, u0, cP , T ) ∀ k = 1, 2, . . . , N (2.73)

and ∥∥∇ukh∥∥2 ≤ C (ε, σ, u0, cP , T ) ∀ k = 1, 2, . . . , N. (2.74)

The inequality (2.72) can be seen as the discrete analogue of (2.28) while (2.73) and

(2.74) correspond respectively to (2.29) and (2.30).

We shall now establish a discrete version of (2.34). We start by writing (2.54) at

time tn and tn−1 and subtracting the two equations that result to get(wnh − wn−1

h

∆t, vh

)= ε2

(∇unh −∇un−1

h

∆t,∇vh

)+

(Φ′(unh)− Φ′(un−1

h )

∆t, vh

),

so setting vh = wnh in this, we have(wnh − wn−1

h

∆t, wnh

)= ε2

(∇unh −∇un−1

h

∆t,∇wnh

)+

(Φ′(unh)− Φ′(un−1

h )

∆t, wnh

).

If we now re-write wnh in the left-most inner product of this last equation as

wnh =wnh − wn−1

h

2+wnh + wn−1

h

2

and rearrange, we obtain

ε2(∇unh −∇un−1

h

∆t,∇wnh

)=

1

2∆t

(‖wnh‖

2 −∥∥wn−1

h

∥∥2)

+1

2∆t

∥∥wnh − wn−1h

∥∥2

−(

Φ′(unh)− Φ′(un−1h )

∆t, wnh

). (2.75)

If we multiply (2.67) by ε2/∆t2, we obtain

ε2∥∥∥∥unh − un−1

h

∆t

∥∥∥∥2

+ ε2(∇wnh ,∇

unh − un−1h

∆t

)+ σε2

(unh −m,

unh − un−1h

∆t

)= 0, (2.76)

25

so then if we substitute (2.75) into (2.76) we see that

ε2∥∥∥∥unh − un−1

h

∆t

∥∥∥∥2

+1

2∆t

(‖wnh‖

2 −∥∥wn−1

h

∥∥2)

+1

2∆t

∥∥wnh − wn−1h

∥∥2

+σε2(unh −m,

unh − un−1h

∆t

)=

(Φ′(unh)− Φ′(un−1

h )

∆t, wnh

). (2.77)

If we now re-write unh −m in the last inner product on the left-hand side of (2.77) as

unh −m =unh − un−1

h

2+

(unh −m) + (un−1h −m)

2

and expand, we determine that

ε2∥∥∥∥unh − un−1

h

∆t

∥∥∥∥2

+1

2∆t

(‖wnh‖

2 −∥∥wn−1

h

∥∥2)

+1

2∆t

∥∥wnh − wn−1h

∥∥2

+σε2

2∆t

(‖unh −m‖

2 −∥∥un−1

h −m∥∥2)

+σε2

2∆t

∥∥unh − un−1h

∥∥2

=

(Φ′(unh)− Φ′(un−1

h )

∆t, wnh

). (2.78)

We focus our attention on the last term in (2.78). Since Φ′(u) = u3 − u, we see that(Φ′(u)− Φ′(v)

∆t, w

)=

((u3 − u)− (v3 − v)

∆t, w

)=

(u− v

∆t

[(u2 + uv + v2)− 1

], w

).

Hence, the right-hand side of (2.78) is(Φ′(unh)− Φ′(un−1

h )

∆t, wnh

)≤

∥∥∥∥unh − un−1h

∆t

∥∥∥∥∥∥[(unh)2 + unhun−1h + (un−1

h )2]wnh∥∥

+

∥∥∥∥unh − un−1h

∆t

∥∥∥∥ ‖wnh‖ (2.79)

by the triangle and Cauchy–Schwarz inequalities in L2(Ω). Now∥∥[(unh)2 + unhun−1h + (un−1

h )2]wnh∥∥2

≤ 3(∥∥(unh)2wnh

∥∥2+∥∥unhun−1

h wnh∥∥2

+∥∥(un−1

h )2wnh∥∥2)

(Cauchy–Schwarz ineq.)

≤ 3(‖unh‖

4L6(Ω) ‖w

nh‖

2L6(Ω) + ‖unh‖

2L6(Ω)

∥∥un−1h

∥∥2

L6(Ω)‖wnh‖

2L6(Ω)

+∥∥un−1

h

∥∥4

L6(Ω)‖wnh‖

2L6(Ω)

)(see Note (1) below)

≤ 3(c6S ‖unh‖

4H1(Ω) ‖w

nh‖

2H1(Ω) + c6

S ‖unh‖2H1(Ω)

∥∥un−1h

∥∥2

H1(Ω)‖wnh‖

2H1(Ω)

26

+c6S

∥∥un−1h

∥∥4

H1(Ω)‖wnh‖

2H1(Ω)

)(see Note (2) below)

= 3c6S

(‖unh‖

4H1(Ω) + ‖unh‖

2H1(Ω)

∥∥un−1h

∥∥2

H1(Ω)+∥∥un−1

h

∥∥4

H1(Ω)

)‖wnh‖

2H1(Ω)

≤ 9c6S

(max

0≤k≤N

∥∥ukh∥∥H1(Ω)

)4

‖wnh‖2H1(Ω) . (2.80)

Notes:

1) We use the Holder inequalities (Theorems 5, 6, p. 56) as follows: firstly,∥∥(unh)2wnh∥∥2 ≤

(∫Ω

(unh)4p dΩ

)1/p(∫Ω

(wnh)2q dΩ

)1/q

= ‖unh‖4L6(Ω) ‖w

nh‖

2L6(Ω) ,

where we set p = 32

and q = 3. Then∥∥unhun−1h wnh

∥∥2 ≤(∫

Ω

(unh)2p dΩ

)1/p(∫Ω

(un−1h )2q dΩ

)1/q (∫Ω

(wnh)2r dΩ

)1/r

= ‖unh‖2L6(Ω)

∥∥un−1h

∥∥2

L6(Ω)‖wnh‖

2L6(Ω) ,

where we set p = q = r = 3. Finally,∥∥(un−1

h )2wnh∥∥2 ≤

∥∥un−1h

∥∥4

L6(Ω)‖wnh‖

2L6(Ω) using

the same reasoning as for the ‖(unh)2wnh‖2

term above, but replacing unh with un−1h .

2) We use Sobolev’s inequality (Theorem 3, p. 55) with ‖unh‖L6(Ω) ≤ cS ‖unh‖H1(Ω) and

‖wnh‖L6(Ω) ≤ cS ‖wnh‖H1(Ω).

Taking square roots on both sides of (2.80) then gives∥∥[(unh)2 + unhun−1h + (un−1

h )2]wnh∥∥ ≤ 3c3

S

(max

0≤k≤N

∥∥ukh∥∥H1(Ω)

)2

‖wnh‖H1(Ω) .

Substituting this back into (2.79) for∥∥[(unh)2 + unhu

n−1h + (un−1

h )2]wnh∥∥ then yields(

Φ′(unh)− Φ′(un−1h )

∆t, wnh

)≤∥∥∥∥unh − un−1

h

∆t

∥∥∥∥‖wnh‖+ 3c3

S

(max

0≤k≤N

∥∥ukh∥∥H1(Ω)

)2

‖wnh‖H1(Ω)

. (2.81)

Noting the triangle inequality and Equation (2.74) we have

max0≤k≤N

∥∥ukh∥∥H1(Ω)≤ max

0≤k≤N

∥∥ukh −m∥∥H1(Ω)+ ‖m‖H1(Ω)

= max0≤k≤N

(∥∥ukh −m∥∥2+∥∥∇ukh∥∥2

)1/2

+ ‖m‖H1(Ω)

≤ max0≤k≤N

(c2P

∥∥∇ukh∥∥2+∥∥∇ukh∥∥2

)1/2

+ ‖m‖H1(Ω)

= (1 + c2P )1/2 max

0≤k≤N

∥∥∇ukh∥∥+m |Ω|1/2

= C(ε, σ, u0,m, cP , T, |Ω|),

27

where we applied Poincare’s inequality (Theorem 8, p. 57) to obtain the third line

and (2.74) in the last step. Hence, from (2.81) we deduce that(Φ′(unh)− Φ′(un−1

h )

∆t, wnh

)≤∥∥∥∥unh − un−1

h

∆t

∥∥∥∥‖wnh‖+ C ‖wnh‖H1(Ω)

,

where we define C := C (ε, σ, u0,m, cP , cS, T, |Ω|) as a shorthand in the forthcoming

manipulations. We substitute this into the right-hand side of (2.78) and obtain

ε2∥∥∥∥unh − un−1

h

∆t

∥∥∥∥2

+1

2∆t

(‖wnh‖

2 −∥∥wn−1

h

∥∥2)

+1

2∆t

∥∥wnh − wn−1h

∥∥2

+σε2

2∆t

(‖unh −m‖

2 −∥∥un−1

h −m∥∥2)

+σε2

2∆t

∥∥unh − un−1h

∥∥2

≤∥∥∥∥unh − un−1

h

∆t

∥∥∥∥‖wnh‖+ C ‖wnh‖H1(Ω)

≤ ε2

2

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+1

2ε2

(‖wnh‖+ C ‖wnh‖H1(Ω)

)2

(Young’s ineq.; Thm. 7, p. 57)

≤ ε2

2

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+1 + C2

2ε2

(‖wnh‖

2 + ‖wnh‖2H1(Ω)

)(Cauchy–Schwarz ineq.)

=ε2

2

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+1 + C2

ε2‖wnh‖

2 +1 + C2

2ε2‖∇wnh‖

2 .

If we rearrange terms, we see that

ε2∥∥∥∥unh − un−1

h

∆t

∥∥∥∥2

+1

∆t

(‖wnh‖

2 −∥∥wn−1

h

∥∥2)

+1

∆t

∥∥wnh − wn−1h

∥∥2

+σε2

∆t

(‖unh −m‖

2 −∥∥un−1

h −m∥∥2)

+σε2

∆t

∥∥unh − un−1h

∥∥2

≤ 2(1 + C2)

ε2‖wnh‖

2 +1 + C2

ε2‖∇wnh‖

2 .

If we sum this through n = 1, 2, . . . k, multiply through by ∆t, and rearrange we get(1− 2(1 + C2)

ε2∆t

)∥∥wkh∥∥2+ ε2

k∑n=1

∆t

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+k∑

n=1

∥∥wnh − wn−1h

∥∥2+ σε2

∥∥ukh −m∥∥2+ σε2

k∑n=1

∥∥unh − un−1h

∥∥2 ≤∥∥w0

h

∥∥2

+ σε2∥∥u0

h −m∥∥2

+ C +2(1 + C2)

ε2

k−1∑n=1

∆t ‖wnh‖2

where we apply (2.73) to∑k

n=1 ∆t ‖∇wnh‖2, and define C = C (ε, σ, u0,m, cP , cS, T, |Ω|)

in the above as the greater of C and the constant C(ε, σ, u0, cP , T ) from (2.73). If we

28

now assume that

1− 2(1 + C2)

ε2∆t ≥ 1

2⇒ ∆t ≤ ε2

4(1 + C2)(2.82)

then we can write

∥∥wkh∥∥2+ 2ε2

k∑n=1

∆t

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+ 2k∑

n=1

∥∥wnh − wn−1h

∥∥2

+ 2σε2∥∥ukh −m∥∥2

+ 2σε2k∑

n=1

∥∥unh − un−1h

∥∥2

≤ 2(∥∥w0

h

∥∥2+ σε2

∥∥u0h −m

∥∥2+ C(ε, σ, u0, cP , T )

)+

4(1 + C2)

ε2

k−1∑n=1

∆t ‖wnh‖2.

The discrete Gronwall Lemma (Theorem 2, p. 55), then allows us to deduce that

∥∥wkh∥∥2+ 2ε2

k∑n=1

∆t

∥∥∥∥unh − un−1h

∆t

∥∥∥∥2

+ 2k∑

n=1

∥∥wnh − wn−1h

∥∥2

+2σε2∥∥ukh −m∥∥2

+ 2σε2k∑

n=1

∥∥unh − un−1h

∥∥2

≤ 2(∥∥w0

h

∥∥2+ σε2

∥∥u0h −m

∥∥2+ C(ε, σ, u0, cP , T )

)exp

(4(1 + C2)

ε2T

)= C(ε, σ, u0,m, cP , cS, T, |Ω|). (2.83)

Consequently, ∥∥wkh∥∥ ≤ C(ε, σ, u0,m, cP , cS, T, |Ω|) ∀ k = 1, 2, . . . , N (2.84)

and ∥∥ukh −m∥∥ ≤ C(ε, σ, u0,m, cP , cS, T, |Ω|) ∀ k = 1, 2, . . . , N. (2.85)

Note: here w0h ∈ Vh is defined by(

w0h, vh

)= −ε2

(∆hu

0h, vh

)+(Φ′(u0h

), vh)∀ vh ∈ Vh,

and therefore, if we set vh = w0h in the above we have∥∥w0

h

∥∥2= −ε2

(∆hu

0h, w

0h

)+(Φ′(u0h

), w0

h

)≤ ε2

∥∥∆hu0h

∥∥∥∥w0h

∥∥+∥∥Φ′

(u0h

)∥∥ ∥∥w0h

∥∥ (Cauchy–Schwarz ineq.)

⇒∥∥w0

h

∥∥ ≤ ε2∥∥∆hu

0h

∥∥+∥∥Φ′

(u0h

)∥∥ .29

However, if we put vh = u0h in (2.58) we get(

u0h, u

0h

)+(∇u0

h,∇u0h

)+(∆hu

0h,∆hu

0h

)=(u0, u

0h

)+(∇u0,∇u0

h

)+(∆Nu0,∆hu

0h

)i.e. if we use the Cauchy–Schwarz inequality repeatedly,∥∥u0

h

∥∥2+∥∥∇u0

h

∥∥2+∥∥∆hu

0h

∥∥2 ≤ ‖u0‖∥∥u0

h

∥∥+ ‖∇u0‖∥∥∇u0

h

∥∥+ ‖∆Nu0‖∥∥∆hu

0h

∥∥≤

(‖u0‖2 + ‖∇u0‖2 + ‖∆Nu0‖2)1/2

×(∥∥u0

h

∥∥2+∥∥∇u0

h

∥∥2+∥∥∆hu

0h

∥∥2)1/2

.

So then ∥∥u0h

∥∥2+∥∥∇u0

h

∥∥2+∥∥∆hu

0h

∥∥2 ≤ ‖u0‖2 + ‖∇u0‖2 + ‖∆Nu0‖2 . (2.86)

Since Φ′(unh) = (unh)3 − unh for n = 0, 1, 2, . . . N , we also have that

‖Φ′(unh)‖ ≤∥∥(unh)3

∥∥+ ‖unh‖

= ‖unh‖3L6(Ω) + ‖unh‖

≤ c3S ‖unh‖

3H1(Ω) + ‖unh‖ (2.87)

by Sobolev’s inequality (Theorem 3, p. 55). Specifically, for n = 0,

∥∥u0h

∥∥3

H1(Ω)=(∥∥u0

h

∥∥2

H1(Ω)

)3/2

≤(‖u0‖2 + ‖∇u0‖2 + ‖∆Nu0‖2)3/2

by (2.86). Substituting for ‖u0h‖

3H1(Ω) in (2.87) we have∥∥Φ′(u0

h)∥∥ ≤ c3

S

(‖u0‖2 + ‖∇u0‖2 + ‖∆Nu0‖2)3/2

+(‖u0‖2 + ‖∇u0‖2 + ‖∆Nu0‖2)1/2

≤ C(u0, cS)

if we apply (2.86) to ‖u0h‖ as well. Hence ‖w0

h‖ ≤ C(ε, u0, cS) and so we were able to

absorb w0h in the constant C appearing on the right-hand side of (2.83).6

Next, we take vh = ∆hunh in (2.54) and obtain

(wnh ,∆hunh) = ε2 (∇unh,∇(∆hu

nh)) + (Φ′(unh),∆hu

nh)

= −ε2 (∆unh,∆hunh) + (Φ′(unh),∆hu

nh)

6See Equation (2.35) for the analogous situation in the continuous case.

30

so then

ε2 ‖∆hunh‖

2 = − (wnh ,∆hunh) + (Φ′(unh),∆hu

nh)

≤ ‖wnh‖ ‖∆hunh‖+ ‖Φ′(unh)‖ ‖∆hu

nh‖

by the Cauchy–Schwarz inequality. Consequently, if we divide through by ‖∆hunh‖,

we are left with

ε2 ‖∆hunh‖ ≤ ‖wnh‖+ ‖Φ′(unh)‖

≤ ‖wnh‖+ c3S ‖unh‖

3H1(Ω) + ‖unh‖

by (2.87). Hence, arguing as on p. 14 in the build-up to Equation (2.39), we see that

max0≤n≤N

‖∆hunh‖ ≤

1

ε2

[max

0≤n≤N‖wnh‖+ c3

S max0≤n≤N

(‖∇unh‖

2 + ‖unh‖2)3/2

+ max0≤n≤N

‖unh‖]

≤ 1

ε2

[max

0≤n≤N‖wnh‖+ c3

S max0≤n≤N

(‖∇unh‖

2 + 2 ‖unh −m‖2

+2 ‖m‖2)3/2+ max

0≤n≤N‖unh −m‖+ ‖m‖

].

Noting (2.74), (2.84) and (2.85) (and (2.86) for the case n = 0) leads to:

‖∆hunh‖ ≤ C(ε, σ, u0,m, cP , cS, T, |Ω|) for n = 0, 1, . . . , N. (2.88)

Assuming that the finite element triangulation is quasi-uniform,7 we also have that∥∥∥∥unh −−∫Ω

unh dΩ

∥∥∥∥L∞(Ω)

≤ C ‖unh‖1−θ ‖∆hu

nh‖

θ

where θ = 1/2 if d = 2, and θ = 3/4 if d = 3 (see [BB99] and [BB01]). Thus we

deduce that

‖unh‖L∞(Ω) ≤ C(ε, σ, u0,m, cP , cS, T, |Ω|) for n = 0, 1, . . . , N. (2.89)

The inequality (2.89) will be crucial in establishing stability in the next section, and

in the error analysis of the method, which we schedule as a future opportunity.8

7A quasi-uniform family of triangulations is one in which each triangulation has the property

that the ratio of the longest side of any triangle therein, h, to the radius of the smallest inscribed

circle is bounded by some constant across the family, as h → 0. (See Defn. 4.4.13 of [BS07].) The

uniform triangulations depicted in Fig. C.1, which we use for our two-dimensional simulations, can

clearly be regarded as members of the same quasi-uniform family.8We note here from [S12] that the error analysis of the finite element method will proceed along

similar lines to the stability analysis in Section 2.3.4, and therefore the bound (2.89) will be crucial,

just as (2.40) was in Section 2.3.4.

31

2.4.4 Stability

We are now in a position to establish a stability relation similar to (2.47) for the finite

element approximation. Following similar reasoning to that noted in Section 2.3.4,

we suppose that the finite element problem has two solutions at time n∆t which we

denote(unh,1, w

nh,1

)and

(unh,2, w

nh,2

). We recall that the physical problem is solved by

u(x, t) and its approximation unh, but that the weak and finite element forms of the

problem have two-part solutions (u(x, t), w(x, t))). Now we define

Unh := (unh, w

nh) =

(unh,1, w

nh,1

)−(unh,2, w

nh,2

)and observe that Un

h solves the problem of finding unh, wnh ∈ Vh such that(

unh − un−1h

∆t, vh

)+ (∇wnh ,∇vh) + σ (unh, vh) = 0 ∀ vh ∈ Vh (2.90)

and

(wnh , vh) = ε2 (∇unh,∇vh) +(Φ′(unh,1)− Φ′(unh,2), vh

)∀ vh ∈ Vh. (2.91)

These equations are obtained by writing Equations (2.53) and (2.54) for(unh,1, w

nh,1

)and

(unh,2, w

nh,2

)and subtracting the results. In (2.90), we set

vh = unh =∆t

2

unh − un−1h

∆t+unh + un−1

h

2

to see that

ε2

2∆t

∥∥unh − un−1h

∥∥2+

ε2

2∆t

(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ ε2 (∇wnh ,∇unh) + ε2σ ‖unh‖2 = 0

(2.92)

after multiplying through by ε2. Then, setting vh = wnh in (2.91) gives

‖wnh‖2 = ε2 (∇unh,∇wnh) +

(Φ′(unh,1)− Φ′(unh,2), wnh

). (2.93)

Adding (2.92) and (2.93) then yields

ε2

2∆t

∥∥unh − un−1h

∥∥2+

ε2

2∆t

(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ ‖wnh‖2 + ε2σ ‖unh‖

2

=(Φ′(unh,1)− Φ′(unh,2), wnh

).

Arguing as we did after (2.45) on p. 16, we know that we can write this last result as

ε2

2∆t

∥∥unh − un−1h

∥∥2+

ε2

2∆t

(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ ‖wnh‖2 + ε2σ ‖unh‖

2

=

∫Ω

unhwnh

[3(θunh,1 + (1− θ)unh,2

)2 − 1]

dΩ

32

for some θ ∈ [0, 1]. We know according to (2.89) that∥∥unh,1∥∥L∞(Ω),∥∥unh,2∥∥L∞(Ω)

≤ C(ε, σ, u0,m, cP , cS, T, |Ω|) for n = 0, 1, . . . , N,

since they are both finite element solutions, and so in the same way as we did fol-

lowing (2.46) on p. 16, we define three constants K(h)1 , K

(h)2 and Ch (corresponding

respectively to K1, K2 and C) and argue that

ε2

2∆t

∥∥unh − un−1h

∥∥2+

ε2

2∆t

(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ 12‖wnh‖

2 + ε2σ ‖unh‖2 ≤ 1

2C2h ‖unh‖

2 .

Then, since∥∥unh − un−1

h

∥∥2 ≥ 0, we can drop this term on the left and write

1

2∆tε2(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ 12‖wnh‖

2 + ε2σ ‖unh‖2 ≤ 1

2C2h ‖unh‖

2 . (2.94)

This is the discrete analogue of Equation (2.47). As in Section 2.3.4, we again have

two cases to consider:

1) ε2σ ≥ 12C2h: in this case, (2.94) gives

ε2(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ ∆t ‖wnh‖2 ≤ 0.

If we sum this through n = 1, 2, . . . k, (k ≤ N) and rearrange, we obtain

ε2∥∥ukh∥∥2

+k∑

n=1

∆t ‖wnh‖2 ≤ ε2

∥∥u0h

∥∥2

for any k = 1, 2, . . . , N , where u0h ≡ u0

h,1 − u0h,2. We conclude that∥∥ukh∥∥ ≤ ∥∥u0

h

∥∥ for any k = 1, 2, . . . , N,

i.e. uh ∈ `∞ (0, T ;L2(Ω)) where we define

‖uh‖`∞(0,T ;L2(Ω)) := max0≤k≤N

∥∥ukh∥∥ ≤ ∥∥u0h

∥∥ <∞. (2.95)

In addition, we have(k∑

n=1

∆t ‖wnh‖2

)1/2

≤ ε∥∥u0

h

∥∥ for any k = 1, 2, . . . , N,

and so if we set the upper limit of summation to k = N (as we may), then

wh ∈ `2 (0, N ;L2(Ω)) where we define

‖wh‖`2(0,T ;L2(Ω)) :=

(N∑n=1

∆t ‖wnh‖2

)1/2

≤ ε∥∥u0

h

∥∥ <∞. (2.96)

33

If we now define the norm of Uh = (uh, wh) ∈ `∞ (0, T ;L2(Ω))× `2 (0, T ;L2(Ω)) in

such a way that

‖Uh‖2`∞(0,T ;L2(Ω))×`2(0,T ;L2(Ω)) := ‖uh‖2

`∞(0,T ;L2(Ω)) + ‖wh‖2`2(0,T ;L2(Ω)) (2.97)

then addition of the squares of (2.95) and (2.96) gives

‖Uh‖`∞(0,T ;L2(Ω))×`2(0,T ;L2(Ω)) ≤√

1 + ε2∥∥u0

h,1 − u0h,2

∥∥ (2.98)

which is the discrete version of (2.51) and means that the finite element approxi-

mation to the problem is stable if ε2σ ≥ 12C2h.

2) ε2σ < 12C2h: in this second case, (2.94) gives

ε2(‖unh‖

2 −∥∥un−1

h

∥∥2)

+ ∆t ‖wnh‖2 ≤ ∆t

(C2h − 2ε2σ

)‖unh‖

2 .

Summing the above through n = 1, 2, . . . k, (k ≤ N) in the usual way and rear-

ranging gives

ε2∥∥ukh∥∥2

+k∑

n=1

∆t ‖wnh‖2 ≤ ε2

∥∥u0h

∥∥2+(C2h − 2ε2σ

) k∑n=1

∆t ‖unh‖2.

We move the k-th term in the series on the right-hand side over to the left-hand

side and obtain

(ε2 −∆t

(C2h − 2ε2σ

))∥∥ukh∥∥2+

k∑n=1

∆t ‖wnh‖2

≤ ε2∥∥u0

h

∥∥2+(C2h − 2ε2σ

) k−1∑n=1

∆t ‖unh‖2.

Assuming(ε2 −∆t

(C2h − 2ε2σ

))≥ ε2

2⇒ ∆t ≤ ε2

2(C2h − 2ε2σ

) (2.99)

then implies that

ε2

2

∥∥ukh∥∥2+

k∑n=1

∆t ‖wnh‖2 ≤ ε2

∥∥u0h

∥∥2

+2(C2h − 2ε2σ

)ε2

k−1∑n=1

∆t

ε2

2‖unh‖

2 +n∑

m=1

∆t ‖wmh ‖2

,

34

where we have strategically added a non-negative term on the right-hand side

in preparation for the application of the discrete Gronwall Lemma (Theorem 2,

p. 55). Using this result, we conclude that

ε2

2

∥∥ukh∥∥2+

k∑n=1

∆t ‖wnh‖2 ≤ ε2C2

h

∥∥u0h

∥∥2,

where we define

Ch :=

√√√√√exp

2(C2h − 2ε2σ

)ε2

n∆t

= Ch(ε, σ, u0

h,1, u0h,2,m, cS, cP , T, |Ω|

).

We notice immediately that∥∥ukh∥∥ ≤ √2Ch∥∥u0

h

∥∥ for any k = 1, 2, . . . , N,

i.e. uh ∈ `∞ (0, T ;L2(Ω)). Moreover,(k∑

n=1

∆t ‖wnh‖2

)1/2

≤ εCh∥∥u0

h

∥∥ for any k = 1, 2, . . . , N,

and so once again, wh ∈ `2 (0, T ;L2(Ω)). Taking the sum of squares as in the Case

(1) proof, we have

‖Uh‖`∞(0,T ;L2(Ω))×`2(0,T ;L2(Ω)) ≤ Ch√

2 + ε2∥∥u0

h,1 − u0h,2

∥∥ (2.100)

which is the discrete version of (2.52) and means that the finite element approxi-

mation to the problem is stable if ε2σ < 12C2h.

We conclude that the sequence of finite element approximations of the problem is

bounded and stable under all circumstances if ∆t satisfies the conditions listed in

(2.82) and (2.99). We notice that these conditions are sufficient (but not necessary)

for boundedness and stability, and that they are also independent of the discretization

parameter h, which measures the fineness of the triangulation.

35

Chapter 3

Implementation

We are now in a position to write down the details related to a specific finite element

implementation of the problem. The matrix form derived below is formally the same

for d = 1, 2, 3 but is easier to imagine with a specific dimension and space Vh in mind.

As a result, we tend to develop the implementation model with one spatial dimension

(d = 1) in mind. Full details related to the Matlab implementation can be seen in

[Par12b].

3.1 A specific numerical scheme

We impose a uniform discretisation on the spatial domain involving M equal subdi-

visions in each direction, assumed constant for all tn, 0 ≤ n ≤ N . We then define

the corresponding set of piecewise linear basis ‘hat’ functions φi in the usual way, for

instance in one dimension as

φi(x) =

(1− |x− xi|

h

)+

for 0 ≤ i ≤ (M + 1)d − 1

with d = 1. Thereafter, we specify the (M + 1)d-dimensional space Vh from the

previous chapter in terms of these basis functions as

Vh = span φi0≤i≤(M+1)d−1 .

In order to find solutions unh and wnh to the finite element problem, we need to write

the coupled problem specified in Equations (2.53) and (2.54) in matrix form. To start,

we express unh and wnh as members of Vh, viz.

unh =

(M+1)d−1∑i=0

Uni φi(x), n = 0, 1, . . . N, (3.1)

36

and

wnh =

(M+1)d−1∑i=0

W ni φi(x), n = 0, 1, . . . N. (3.2)

Notice that we can identify the (approximate) solutions unh and wnh at time tn = n∆t

respectively with the vectors Un (with components Uni ) and W n (with components

W ni ) in R(M+1)d−1.

Instead of taking the full H2-projection of some function u0 ∈ H2(Ω) onto Vh (see

Section 2.4.1), we specify an initial condition ‘on the grid’ of the form

u0(x) =

(M+1)d−1∑i=0

Riφi(x)

and then require that the following simplified version of Equation (2.58) holds:(u0h, vh

)= (u0, vh) for all vh ∈ Vh,

that is, (M+1)d−1∑i=0

U0i φi, vh

=

(M+1)d−1∑i=0

R0iφi, vh

for all vh ∈ Vh.

Writing the equation above for each φj in the finite-dimensional basis of Vh and using

linearity leads to a system of (M + 1)d equations for the initial condition as follows:

(M+1)d−1∑i=0

U0i (φi, φj) =

(M+1)d−1∑i=0

R0i (φi, φj) (3.3)

for 0 ≤ j ≤ (M + 1)d − 1. Proceeding in the normal way, we now define a ‘mass

matrix’

M = (mij) where mij = (φj, φi) =

∫Ω

φjφi dΩ (3.4)

and a ‘stiffness matrix’

S = (sij) where sij = (∇φj,∇φi) =

∫Ω

∇φj · ∇φi dΩ. (3.5)

Using these, it is clear that an equivalent matrix problem to that in Equation (3.3) is

MU0 = MR

37

or, if M is invertible (which it is),

U0 = M−1MR = IR = R. (3.6)

We consider Equations (2.53) and (2.54). As with the initial condition, we write these

equations for each φj in the basis of Vh, use linearity and obtain the coupled system

(1 + σ∆t)

(M+1)d−1∑i=0

Uni (φi, φj) + ∆t

(M+1)d−1∑i=0

W ni (∇φi,∇φj)

=

(M+1)d−1∑i=0

Un−1i (φi, φj) + σ∆t (m,φj) (3.7)

and

(M+1)d−1∑i=0

W ni (φi, φj) = ε2

(M+1)d−1∑i=0

Uni (∇φi,∇φj) +

Φ′

(M+1)d−1∑i=0

Uni φi

, φj

(3.8)

for 0 ≤ j ≤ (M + 1)d − 1. We see that we have a significant non-linearity in the last

term on the right-hand side of (3.8). To address this, we define an iterative scheme

for each time step as follows: we know that Φ′(u) = u(u2− 1) and so we approximateΦ′

(M+1)d−1∑i=0

Uni φi

, φj

≈

((M+1)d−1∑

i=0

Un,ki φi︸ ︷︷ ︸

‘u’

(M+1)d−1∑l=0

Un,k−1l φl

2

− 1

︸ ︷︷ ︸

‘u2 − 1’

, φj

)

=

(M+1)d−1∑i=0

Un,ki

∫Ω

(M+1)d−1∑l=0

Un,k−1l φl

2

φiφj dΩ

−

(M+1)d−1∑i=0

Un,ki

(∫Ω

φiφj dΩ

)using linearity, for each j, 0 ≤ j ≤ (M + 1)d − 1. Now we define the matrix

L(n,k) =(l(n,k)ij

)where l

(n,k)ij =

∫Ω

(M+1)d−1∑l=0

Un,k−1l φl

2

φjφi dΩ (3.9)

and write the coupled system in (3.7) and (3.8) as an iterative scheme using (3.4),

(3.5) and (3.9) as

(1 + σ∆t)MU (n,k) + ∆tSW (n,k) = F (n) (3.10)

38

with (−ε2S + M − L(n,k)

)U (n,k) + MW (n,k) = 0 (3.11)

where the vector F (n) =(F

(n)0 , F

(n)1 , . . . , F

(n)

(M+1)d−1

)Tis defined component-wise via

F(n)j =

(MUn−1

)j

+ σ∆t (m,φj) for 0 ≤ j ≤ (M + 1)d − 1. (3.12)

We summarise the scheme as((1 + σ∆t)M ∆tS

−ε2S + M − L(n,k) M

)(U (n,k)

W (n,k)

)=

(F (n)

0

)(3.13)

where we define

Un := limk→∞

U (n,k) and W n := limk→∞

W (n,k) (3.14)

and specify the starting condition at each time level as the ending condition from the

previous one, that is to say, we set U (n,0) := Un−1.

We summarise the high level algorithm for arbitrary spatial dimensions as follows:

1) We solve (3.6) to find U0, the projection of the initial condition onto Vh.1

2) We then define U (1,0) = U0 and use this in Equation (3.9) to build the matrix

L(1,1) and in Equation (3.12) to build the vector F (1).

3) We solve the matrix problem (3.13) to find U (1,1) and W (1,1) from((1 + σ∆t)M ∆tS

−ε2S + M − L(1,1) M

)(U (1,1)

W (1,1)

)=

(F (1)

0

).

4) U (1,1) is then used in Equation (3.9) to build the matrix L(1,2). The matrix problem

(3.13) is then solved to find U (1,2) and W (1,2). The process is repeated until sub-

sequent vector pairs(U (1,k), U (1,k−1)

)are considered sufficiently close2 as to have

‘converged’. The result is defined as the solution (U1,W 1) at the first time-step.

1In practice, we wish to specify a precise initial mass, but observe that a set of appropriately

distributed random numbers typically has a mass that is only close to what we want. We achieve fine

control by managing the value of the initial condition on one node in two dimensions (see Section

1.5 of [Par12b] for details). Hence, we speak of ‘all-but pseudo-random’ initial conditions.2In the code, we use a stopping condition of the following form to terminate the process:∥∥∥U (1,k) − U (1,k−1)

∥∥∥∞< TOL.

39

5) We repeat the process above at subsequent time-steps, increasing n by one at each

step, until we have n = N .

3.2 Results summary: two dimensional space

Our aim in this section is to use our Matlab implementation to examine the effect

on the free-energy evolution and end-state morphology, of varying the initial system

mass, m, and the non-local energy coefficient, σ, in two spatial dimensions.3

3.2.1 The effect of varying mass

Our exploration of the effects of varying the system mass is based on the parameters

depicted in Table 3.1.

Type Parameter Values

Physical Mass (m) 0.0 0.4

ε 0.08

σ 10

Control ∆t (:= ε2) 0.0064

Time Steps (N) 2 000

Total Simulated Time (T ) 12.8 secs

h (in x and y) 1/20

Number of Elements 20× 20× 2 = 800

Number of Nodes 21× 21

Max Non-Linear Iterations 500

Non-Linear Tolerance (TOL) 10−9

Table 3.1: Physical and control parameters – varying mass in two dimensions

Some remarks in respect of these parameter values are in order:

• We consider two cases: the symmetric case m = 0 (in which the polymers have

an equal number of type A and B monomers) and the non-symmetric cases with

3A more fundamental set of Matlab results and analysis including those relating to the one-

dimensional case can be found in [Par12a].

40

m = 0.4 which corresponds to a situation where we have 70% type A and 30%

type B monomers on each copolymer molecule (see Equation (1.2)).

• The remaining physical parameters are not chosen to be particularly physically

meaningful and are based, in part, on parameters used in [Zha06] and our

experiences reported in [Par12a]. Note that we do not take Equations (1.2),

(1.3) and (1.4), into account in compiling the list of parameters used in this

section (or the next one). Our interest here is in the end-values of m, ε and σ

and not how they, in turn, depend on more fundamental quantities.

• Much experimentation went into selecting the values of N noted in Tables 3.1

and 3.2. We note here that setting N = 2 000 seems to offer a fair balance

between program runtime and the requirement that N be large enough for the

simulated system to achieve what we tentatively term ‘metastability’.4

• As noted in [Par12a], we find that setting ∆t = ε2 does not appear to impact

stability or boundedness but it does allow us to achieve longer simulated time

horizons than would have been the case had we meticulously applied the suf-

ficient boundedness and stability conditions specified in inequalities (2.82) and

(2.99).

• The spatial discretisation used for these experiments is relatively coarse (see

Fig. C.1(a), p. 63) but is based on experimentation with reasonable performance

and system runtime. Additionally, in comparing the effect of varying mass, we

do not anticipate much variation in the fine structure of our end-state solutions.

We turn our attention now to a brief discussion of the results that were observed.5

Given that we are varying the system mass, we anticipate significantly different end-

state solutions for the different mass scenarios and so we focus on these.

The two distinct, random initial conditions (we need to start with two different

system masses) are depicted in Fig. 3.1. We can ‘eyeball’ these distinct masses if

we look at the ranges of values expressed in the horizontal colour bars beneath each

graph; Fig. 3.1(b) clearly represents a non-physical initial condition in which |u0h| > 1

at several points yet the method is robust enough for a physically reasonable end-state

(∣∣uNh ∣∣ < 1 everywhere) to emerge at the end.

4We use this term as we have no way of knowing whether lower energies may be achieved at much

later times.5Animations are available on-line as follows: the m = 0.0 simulation can be viewed at

http://youtu.be/bRSco2N018k, and m = 0.4 at http://youtu.be/p95Q8C1o9sU.

41

x

y

ε=0.08, σ=10 and m=0 at t=0

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5

(a) m = 0

x

y

ε=0.08, σ=10 and m=0.4 at t=0

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5 1

(b) m = 0.4

Figure 3.1: Typical initial conditions (2D)

The evolution pattern observed in most of the simulations is the same: from

a highly variable initial condition, the system quickly moves to a state in which

unh ≈ m after which periodic structures slowly emerge. Our end-state ‘metastable’

solutions are as expected: the symmetric case (m = 0) in Fig. 3.2(a) settles into

a pattern of approximately straight lines of alternating patches of uNh = ±1, while

the non-symmetric case in Fig. 3.2(b) settles into a pattern of repeating circles of

uNh = −1. An informal discussion of the likely stability characteristics of these end-

state solutions is deferred to Appendix C.1.1. The zero Neumann boundary condition

on uNh is clearly satisfied as the contour lines are perpendicular to ∂Ω.

x

y

ε=0.08, σ=10 and m=0 at t=12.8

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5

(a) m = 0

x

y

ε=0.08, σ=10 and m=0.4 at t=12.8

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5

(b) m=0.4

Figure 3.2: Typical ‘metastable’ end-state solutions (2D) for various m

Finally, we notice from Fig. 3.3 that total free energy is a non-increasing function

42

of time irrespective of the system mass, as expected. The end-state morphologies

(lamellar, circular) that result do so because of the interaction and competition be-

tween the three components of the free energy.

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(a) m = 0

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(b) m = 0.4

Figure 3.3: Typical total free energy evolution over time (2D)

In two dimensions, we put Ω := (0, 1)2 and write the free energy functional6 as

E(unh) =

∫(0,1)2

ε2

2|∇unh|

2︸ ︷︷ ︸Part 1

+ 14

(1− (unh)2

)2︸ ︷︷ ︸Part 2

dxdy +σ

2‖unh −m‖

2H−1(0,1)2︸ ︷︷ ︸

Part 3

.

The ‘Part 1’ contribution to the free energy7 is minimised when |∇unh| = 0 i.e. unhis constant and/or the regions on which |∇unh| 6= 0 are as small as possible. It is

clear from Fig. 3.2 that the system has evolved to a state where∣∣∇uNh ∣∣ ≈ 0 whenever

uNh ≈ ±1 over large portions of the space that are coloured black/white. The internal

layers, on which∣∣∇uNh ∣∣ is large, are also visibly narrow as these penalise (i.e. increase)

the contribution of this part of the energy.

The ‘Part 2’ contribution to the free energy is minimised when the double-well

potential function (1− (unh)2)2

is minimised i.e. when unh ≈ ±1. It is almost as if the

‘Part 1’ contribution wants unh to be constant and the ‘Part 2’ contribution specifies

the value of this constant to be ±1. This is clear from Fig. 3.2.

The minimisation of the ‘Part 3’ contribution to the free energy is expressed in

the patterns that emerge. This part of the energy favours a solution in which unh ≈ m

as often as possible, but since energy is the sum of this and two other components

(that together favour unh = ±1 wherever possible), the effect of this nonlocal8 term

6Appendix C.1.2 includes a graphical view of the evolution of the three free energy components

for these experiments.7What we have labeled ‘Part 1’, ‘Part 2’ and ‘Part 3’ are called the ‘Interfacial’, ‘Double-Well’

and ‘Nonlocal Interaction’ energies, respectively, in [Zha06].8. . . so-called because it involves m which is a global parameter, and an integral over (0, 1)2.

43

is to introduce oscillations into the end-state solutions. Numerical experiments and

ansatz-driven analysis (see [CMW11]) predict that in two dimensions, when m = 0

we can expect linear structures to emerge that have low energy (as in Fig. 3.2(a))

and when m 6= 0, we can expect curved (circular) structures (as in Fig. 3.2(b)).

3.2.2 The effect of varying the non-local energy coefficient

In the previous section, we examined the effect of the three components of the free

energy on pattern formation and observed that whereas the ‘Part 1’ and ‘Part 2’

components are minimised when unh = ±1, the ‘Part 3’ component is minimised when

unh = m. The resulting competition between these components is what results in the

regular patterns that emerge. In this section, we explore the effects of varying the

non-local (‘Part 3’) energy coefficient, σ, based on the parameters presented in Table

3.2. For these four experiments, we set m = 0.4, hold ε constant at 0.02 and make

use of the same random initial condition (see Fig. C.4, p. 66).

Type Parameter Values

Physical Mass (m) 0.4

ε 0.02

σ 2 20 200 800

Control ∆t (:= ε2) 0.0004

Time Steps (N) 3 000 2 000 146

Total Simulated Time (T ) 1.2 secs 0.8 secs 0.06 secs

h (in x and y) 1/30

Number of Elements 30× 30× 2 = 1 800

Number of Nodes 31× 31

Max Non-Linear Iterations 500

Non-Linear Tolerance (TOL) 10−9

Table 3.2: Physical and control parameters – varying σ in two dimensions

We are mostly interested here in the effect on solution periodicity of the param-

eter σ and anticipate the effect depicted in Fig. 3.4 i.e. smaller structures (increased

periodicity) as σ increases. We expect the underlying cause of this effect to manifest

itself in the way the components of the free energy interact and on the size of the

44

end-state structures9 that result. We also make use of a somewhat refined (uniform,

30 × 30) mesh for these experiments in hopeful anticipation of some distinctive fine

structure in the results (see Fig. C.1(b), p. 63).

Figure 3.4: SCMFT context

As expected, we observe in Fig. 3.510

that smaller, ‘more periodic’ structures

generally result from larger values of σ.

The reason we associate increasing σ

(for constant ε) with a move ‘down’ the

SCMFT phase diagram in Fig. 3.4 has

to do with Equations (1.3) and (1.4)

where it is clear that increasing σ is

associated with decreasing NP (smaller

molecules) which in turn decreases the

product χNP . In turn, the way this

change in σ causes these effects becomes clear when we examine how the three com-

ponents of the free energy11 relate over time for various σ in Fig. 3.6:

(a) For σ = 2, Fig. 3.5(a) reveals that the system evolves into a state with a small

number of relatively large structures per unit ‘cube’.12 Fig. 3.6(a) reveals that

with this small value of σ, the ‘Part 3’ energy is dominated by the other two parts

so that its impact on structure formation is minimal.

(b) As σ is increased to σ = 20 (i.e. by a factor of ten), Fig. 3.5(b) shows an end-

9An informal discussion of the likely stability characteristics of these end-state solutions is deferred

to Appendix C.2.2.10Animations for the four experiments are available on-line at http://youtu.be/VIH9cNrG8JQ,

http://youtu.be/qd4CXzlI9Bk, http://youtu.be/z2BlRbPcpjk, and http://youtu.be/2z0K8-9IKxI

respectively.11Appendix C.2.3 includes a graphical view of the evolution of the total free energy for these

experiments.12Empirical evidence suggests that the smaller the value of σ, the longer it takes for the system

to become stable and this experiment was no exception. The structures in Fig. 3.5(a) are clearly

non-uniform, and the total energy evolution graph in Fig. C.6(a), p. 68 shows a marked decline in

energy relatively late in time. This explains why the value chosen for N in Table 3.2 was higher for

this experiment than for the others.

To explore this further, the end-state in Fig. 3.5(a) was subsequently used as the initial condition

for a supplementary simulation of another N = 2 000 steps. This extended the total system runtime

for this experiment from ∼40 hours to ∼60 hours. The results of this extended run are visually

virtually indistinguishable from those in Fig. 3.5(a). Nevertheless, they are included in Appendix

C.2.4 on p. 68.

45

state solution that includes approximately twice as many structures as we saw for

σ = 2. This is consistent with the prediction in [Zha06] that to double periodicity,

we should increase σ by a factor of approximately eight. If we look at Fig. 3.6(b),

we find that the corresponding ‘Part 3’ energy is proportionately much larger (by

a factor of approximately ten) which we expect for the larger value of σ. In this

case, the effect is richer structure to keep this ‘Part 3’ contribution small. As

a result, the ‘Part 2’ energy approximately doubles as the number of structures

and the volume occupied by internal layers increases.

(c) As we increase σ by another factor of ten to σ = 200 we see the number of

structures in Fig. 3.5(c) approximately doubles again – as expected. In Fig. 3.6(c),

we observe little change in the actual ‘Part 3’ energy compared to Fig. 3.6(b) as

the finer structure keeps its contribution more or less steady. However, we do

see that the ‘Part 2’ energy has approximately doubled again, as the number of

structures and the volume occupied by internal layers increases further.13

(d) The disordered state14 depicted in Fig. 3.5(d) is especially interesting (as is its

animation) as σ is so large in this case that even small deviations from m lead

to the ‘Part 3’ energy dominating the other two components. Consequently, the

system is pushed to a state with uNh ≈ m everywhere.15 As a result, Fig. 3.6(d)

shows that the ‘Part 3’ energy is essentially zero precisely because the solution

tends to ‘disorder’. This result is consistent with Theorem 3.1 of [CPW09] which

predicts that “u ≡ m is the unique global minimiser. . . if 1 −m2 ≤ 2ε√σ.” We

also see that the ‘Part 1’ energy (favouring constant solutions) tends to zero,

while the ‘Part 2’ energy (favouring unh ≈ ±1) increases even further.

We conclude with an observation in respect of the ‘Part 1’ energy: in all cases in

Fig. 3.6, this initially drops dramatically and then increases. This is consistent with

our earlier observation of typical system evolution: the system initially moves to a

13We use this experiment to start examining h-independence in Appendix C.2.5, p. 70. The idea

is to run 500 steps of this experiment on a 100× 100 mesh.14The term ‘disorder’ has physical significance in the literature. It describes the way in which the

copolymer molecules do not exhibit any regular structure at large times.15Fig. 3.5(d) essentially depicts machine noise about an average of u = 0.4 – the range of values

depicted (i.e. the difference between the largest and smallest components of uNh ) is 4.7×10−15. Any

colour variation in the figure is therefore somewhat meaningless and in some sense, the depicted

solution can be regarded as anything but numerically disordered. In addition, this end-state emerged

extremely quickly: after 146 time-steps, the step-by-step solution change was less than machine

epsilon which is why we selected N = 146 for this experiment in Table 3.2.

46

state with unh close to m, after which structures emerge. We therefore expect |∇unh|and the ‘Part 1’ energy to be small briefly, while unh ≈ m.

x

y

ε=0.02, σ=2 and m=0.4 at t=1.2

0 0.5 10

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5(a) σ = 2

x

y

ε=0.02, σ=20 and m=0.4 at t=0.8

0 0.5 10

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5(b) σ = 20

x

y

ε=0.02, σ=200 and m=0.4 at t=0.8

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5(c) σ = 200

x

y

ε=0.02, σ=800 and m=0.4 at t=0.0584

0 0.5 10

0.2

0.4

0.6

0.8

1

0.4 0.4 0.4 0.4 0.4(d) σ = 800 (‘disorder’)

Figure 3.5: End-state solutions (2D) for various σ

47

0 0.5 10

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 0.5 10

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Part 2

0 0.5 10

2

4

6

8x 10

−3

time

Fre

e E

nerg

y

Part 3

(a) σ = 2

0 0.50

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 0.50

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

yPart 2

0 0.50

0.005

0.01

0.015

timeF

ree

Ene

rgy

Part 3

(b) σ = 20

0 0.50

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 0.50

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Part 2

0 0.50

0.01

0.02

time

Fre

e E

nerg

y

Part 3

(c) σ = 200

0 0.050

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 0.050

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Part 2

0 0.050

0.02

0.04

time

Fre

e E

nerg

y

Part 3

(d) σ = 800 (‘disorder’)

Figure 3.6: Competing free energy components (2D) for various σ. The red circles

depict the component energy levels of the initial condition.

48

Chapter 4

Conclusion

The project made a useful start to the analysis of the OKDE and its finite element ap-

proximation. Theoretical boundedness and stability results were established in both

the analytic and approximate cases and the finite element method was implemented

in one and two spatial dimensions in Matlab. Numerical results agreed with phys-

ical expectation and results obtained by other researchers using alternate (spectral)

methods (see for instance [CMW11] and [Zha06]). They are also consistent with the

theoretical boundedness and stability results we established.

4.1 Opportunities for further study

Several pieces of theoretical analysis could be explored including the existence and

uniqueness of solutions of the OKDE and its finite element approximation. A related

analysis of local and global functional minimisers would be useful and might consider,

inter alia questions such as: can we only expect local, metastable energy minimisers

to emerge or does the OKDE lead to a (unique?) global minimiser? If stable, global

minimisers do exist, are they always numerically accessible? Can we guarantee that

(estimated) free energy is a non-increasing function of time in the finite element

approximation? Many open questions exist in this area; the paper by Choksi et al

([CPW09]) gives an overview of some of the current thinking on these issues.

An analysis of the error and convergence characteristics of the finite element

method would also be useful and should be possible within the framework established

by the stability argument presented in this manuscript. In addition, the impact of any

variational crimes should be understood. The most obvious targets for this analysis

include the integration approximations and the non-linear fixed point iteration that

49

we implemented in Matlab.

The biggest shortcoming of our implementation was that it took several hours to

simulate relatively short time horizons.1 A number of technical options were iden-

tified in this regard, including the use of an alternate spatial discretisation and/or

p-adaptivity. Such measures could allow a more accurate solution to be represented

by a smaller set of data.2 Additionally, the fixed point iteration we used to deal with

the non-linear portion of the problem was computationally expensive. An alterna-

tive involving a variant of Newton’s method might be possible which would likely

substantially reduce overall computational cost. Several opportunities are also noted

in respect of upgrading the underlying linear system solvers (conjugate gradients,

GMRES, multigrid etc.) as well as the integration modules. Bespoke three-point

Gaussian quadrature routines seemed a pragmatic choice when the implementation

was designed, but it is possible that alternatives could reduce runtime and improve

accuracy. Moreover, an implementation in a compiled language such as C/C++ or

FORTRAN could speed up processing (in preparation for three spatial dimensions).

Fundamental implementation and modelling enhancements are also possible. The

most obvious, pressing and interesting enhancement is a three-dimensional spatial

implementation. Such a code would allow for an exploration of the full set of ex-

perimentally observed, physically relevant and complex morphologies illustrated in

Fig. 1.2 on p. 4.

Finally, we observe that all of the analysis performed thus far relates to diblock

copolymers in the pure melt. There is also significant industrial interest in block

copolymer solutions (dilute and concentrated) and it would be interesting to study

these. The key complexity anticipated in this regard entails coupling the system

studied in this document with the Navier Stokes equations for a viscous solvent in

some way. To an extent, this would represent a fundamentally new model which would

require its own analysis in respect of solution existence, uniqueness, boundedness and

stability. This work will be the author’s primary focus for the next few years.

1For ε = 0.02 and σ = 2 in two dimensions for instance, it took ∼40 hours to simulate a mere

1.2 seconds (N = 3 000), whereas we would have preferred to simulate 200+ seconds! This example

is quite extreme but it illustrates the general point of the performance intensity of the calculations

we performed.2h-Adaptivity would add significant analytical complexity.

50

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53

Appendix A

Useful Mathematical Results

Note: several of the results that follow are drawn from [Eva98], and as such, apply to

regions with C1 and C2 boundaries. Given that we work on a unit d-cube Ω = (0, 1)d

in the main text, a more appropriate source for these results would be [Ada75]. This

latter, more advanced work quotes analogous results for Lipschitz domains (i.e. do-

mains that are C1 or C2 a.e.).

A.1 Gronwall’s Lemma

We quote a useful theorem (from [AW09]):

Theorem 1. (Gronwall – continuous) Suppose f is a continuous function on [a, b]

which satisfies

f(t) ≤ g(t) +

∫ t

a

h(s)f(s) ds, t ∈ [a, b]

where g is continuous, h ∈ L1(a, b) and h(t) ≥ 0 a.e.. Then

f(t) ≤ g(t) +

∫ t

a

g(s)h(s) exp

(∫ t

s

h(τ) dτ

)ds for all t ∈ [a, b].

In addition, if g is nondecreasing, then

f(t) ≤ g(t) exp

(∫ t

a

h(s) ds

)for all t ∈ [a, b].

In the case where h(s) = c > 0, these inequalities reduce to:

f(t) ≤ g(t) + c

∫ t

a

g(s)ec(t−s) ds for all t ∈ [a, b].

54

and

f(t) ≤ g(t)ec(t−a) for all t ∈ [a, b].

The discrete counterpart to this, drawn from a result in [S10], is:

Theorem 2. (Gronwall – discrete) If

|ek| ≤ K∗ + L∗k−1∑n=0

∆t|en|, n = 1, 2, . . . N

then

|ek| ≤ K∗eL∗n∆t, n = 1, 2, . . . N.

The notation in this last theorem is drawn from the context of error analysis (as

this was its application in [S10]); but the result is nonetheless formally clear and

applied directly in this paper.

A.2 Sobolev inequalities

Combining Theorem 2 from Section 5.6 of [Eva98] and [S12], we have

Theorem 3. (Sobolev’s inequality) If Ω is a bounded, open subset of Rd with a C1

boundary ∂Ω, and u ∈ W 1,p(Ω) then

‖u‖Lp(Ω) ≤ cS ‖u‖H1(Ω) , 1 ≤ p <∞ for d = 1, 2 (A.1)

and

‖u‖L

2dd−2 (Ω)

≤ cS ‖u‖H1(Ω) , for d > 2.

Specifically, for d = 1, 2, 3,

‖u‖L6(Ω) ≤ cS ‖u‖H1(Ω) . (A.2)

We now quote Theorem 6, part (ii) of Section 5.6 of [Eva98] with k = 2, p = 2,

d = 1, 2, 3:

Theorem 4. (Sobolev embedding) Let Ω be a bounded, open subset of Rd, with a C1

boundary and suppose that u ∈ H2(Ω). If

d < 4

55

then

u ∈ C0,γ(Ω),

where

γ =

12, if d = 1, 3;

any positive number < 1, if d = 2.

Additionally,

‖u‖C0,γ(Ω) ≤ C ‖u‖H2(Ω) ,

where C = C(d, |Ω|), and since C0,γ(Ω) is embedded in L∞(Ω),

‖u‖L∞(Ω) ≤ cS ‖u‖H2(Ω) .

A.3 The Holder inequalities

We quote Proposition 3.3.2 from Section 3 of [Coh80]:

Theorem 5. (Holder’s inequality) Let p and q satisfying 1 ≤ p, q ≤ +∞ be conjugate

exponents i.e.

1

p+

1

q= 1.

If f ∈ Lp(Ω) and g ∈ Lq(Ω), then fg ∈ L1(Ω) and∫Ω

fg dΩ ≤ ‖f‖Lp(Ω) ‖g‖Lq(Ω) . (A.3)

A general form of this theorem (from Result (g) of Appendix B.2 of [Eva98]) is

the following:

Theorem 6. (General Holder inequality) Let 1 ≤ p1, . . . , pm ≤ ∞ with

1

p1

+ . . .+1

pm= 1

and uk ∈ Lpk(Ω) for k = 1, . . . ,m. Then∫Ω

|u1 · · · um| dΩ ≤m∏k=1

‖uk‖Lpk (Ω). (A.4)

56

A.4 Other useful identities and inequalities

We start with a version of Young’s inequality (see [Eva98], Result (c) of Appendix

B.2).

Theorem 7. (Young’s inequality with p = q = 1/2)

ab ≤ 1

2γa2 +

γ

2b2 for any γ > 0. (A.5)

Setting γ = 1 in Equation (A.5) leads to

ab ≤ a2

2+b2

2

whence

2ab ≤ a2 + b2

and so

a2 + 2ab+ b2 ≤ 2(a2 + b2),

that is

(a+ b)2 ≤ 2(a2 + b2). (A.6)

Next we quote Theorem 1 from Section 5.8 of [Eva98]:

Theorem 8. (Poincare’s inequality) Let Ω be a bounded, connected, open subset of

Rd with a C1 boundary ∂Ω and suppose 1 ≤ p ≤ ∞. Then there exists a constant cP ,

depending only on d, p and Ω such that∥∥∥∥v −−∫Ω

v dΩ

∥∥∥∥Lp(Ω)

≤ cP ‖∇v‖Lp(Ω)

for each function v ∈ H1(Ω). Specifically, for p = 2 as in the main body of this paper,

we use ∥∥∥∥v −−∫Ω

v dΩ

∥∥∥∥ ≤ cP ‖∇v‖ . (A.7)

Finally, we apply the Divergence Theorem to manipulate inner products involving

functions in H2(Ω) that have a zero Neumann boundary condition:

57

Theorem 9. (A useful integral identity) If w ∈ H2(Ω) and

∂w

∂n

∣∣∣∣∂Ω

= 0

then ∫Ω

∇v · ∇w dΩ = −∫

Ω

v∆w dΩ (A.8)

for all v ∈ H1(Ω).

Proof. We have

∇ · (v∇w)) = ∇v · ∇w + v∆w

so ∫Ω

v∆w dΩ =

∫Ω

∇ · (v∇w)) dΩ−∫

Ω

∇v · ∇w dΩ

=

∫∂Ω

v∇w · n ds︸ ︷︷ ︸Divergence Theorem

−∫

Ω

∇v · ∇w dΩ

=

∫∂Ω

v∂w

∂nds︸ ︷︷ ︸

= 0 by the BC

−∫

Ω

∇v · ∇w dΩ

= − (∇w,∇v)

and we are done.

We combine Theorem 8.12 of Section 8.4 of [GT01] with Theorem 4.3.1.4 of [Gri85]

and some remarks in [S12]. The theorem from [GT01] applies to a convex polygon

Ω in R2 while the result in [Gri85] applies to Ω, a subset of Rd with a boundary ∂Ω

that is of class C2. Together, these theorems imply the following:

Theorem 10. Under the conditions noted above in respect of Ω, there is a positive

constant C, independent of u such that

‖u‖H2(Ω) ≤ C ‖∆u‖ . (A.9)

58

Appendix B

Calculation Essentials

This chapter contains details of some of the longer calculations that are abbreviated

in the main body of the text. These sections only make sense with reference to those

parts of Chapter 2 whence they are referenced.

B.1 Derivation of the dynamic equation

Continuing the calculation from Equation (2.11) on p. 7, we see that

〈DE(u), φ〉 = limα→0

1

2α

∫Ω

ε2(|∇u|2 + 2α(∇u · ∇φ) + α2|∇φ|2

)+

1

2

(1− u2 − αφ(2u+ αφ)

)2

+σ[(−∆N)−1

((u−m) + αφ

)][(u−m) + αφ]

−ε2 |∇u|2 − 12

(1− u2

)2

− σ[(−∆N)−1

(u−m

)][u−m]

dΩ

(where we used (2.7) twice)

= limα→0

1

2α

∫Ω

ε2(|∇u|2 + 2α(∇u · ∇φ) + α2|∇φ|2

)+

1

2

(1− u2

)2 − αφ(1− u2

)(2u+ αφ) +

1

2α2φ2 (2u+ αφ)2

+σ[(−∆N)−1

(u−m

)+ α(−∆N)−1φ

][(u−m) + αφ]

−ε2 |∇u|2 − 12

(1− u2

)2

− σ[(−∆N)−1

(u−m

)][u−m]

dΩ

(where we used the fact that (−∆N)−1 is a linear operator)

59

= limα→0

1

2α

∫Ω

ε2(2α(∇u · ∇φ) + α2|∇φ|2

)−αφ

(1− u2

)(2u+ αφ) + 1

2α2φ2 (2u+ αφ)2

+ ασφ[(−∆N)−1

(u−m

)]+ ασ [(u−m) + αφ] (−∆N)−1φ

dΩ

= limα→0

∫Ω

ε2(

(∇u · ∇φ) +α

2|∇φ|2

)−1

2φ(1− u2

)(2u+ αφ) + 1

4αφ2 (2u+ αφ)2

+ 12σφ[(−∆N)−1

(u−m

)]+ 1

2σ [(u−m) + αφ] (−∆N)−1φ

dΩ

=

∫Ω

ε2 (∇u · ∇φ)− φu

(1− u2

) dΩ

+σ

2

∫Ω

φ[(−∆N)−1

(u−m

)]+ [u−m] (−∆N)−1φ

dΩ.

We focus on the last line and write it as

σ

2

(φ, (−∆N)−1

(u−m

))+((−∆N)−1φ, u−m

) =σ

2

(φ, (−∆N)−1

(u−m

))+(φ, (−∆N)−1

(u−m

)) by (2.6) and conclude that

〈DE(u), φ〉 = ε2∫

Ω

(∇u · ∇φ) dΩ−∫

Ω

φu(1− u2

)− σφ(−∆N)−1

(u−m

)dΩ

= −ε2∫

Ω

φ∆Nu dΩ−∫

Ω

φu(1− u2

)− σφ(−∆N)−1

(u−m

)dΩ

after applying Theorem 9 (Appendix A.4, p. 58) to the first integral on the right

above and observing the zero Neumann boundary condition on u (note that we do

not require a zero Neumann boundary condition of φ (or more generally, the space

H1∗ (Ω))). We now define

N (u) = u(u2 − 1

),

which is Equation (2.12), to arrive at

〈DE(u), φ〉 = −∫

Ω

φε2∆Nu−N (u)− σ(−∆N)−1

(u−m

)dΩ,

which is Equation (2.13).

60

B.2 Approach and results highlights: the weak

form

Key results are numbered in Fig. B.1 as in the main body of the text in red.

Figure B.1: Argument summary – weak form boundedness and stability

61

B.3 Approach and results highlights: the finite el-

ement approximation

Key results are numbered in Fig. B.2 as in the main body of the text in red.

Figure B.2: Argument summary – finite element approximation boundedness and

stability

62

Appendix C

Supplementary Results and Graphs

The graphs and results presented here are supplementary to those presented in Section

3.2 of the main text. We start with a view of the two two-dimensional discretisations

that were used: the 20× 20 grid used in Section 3.2.1 (see Fig. C.1(a)) and another,

finer 30× 30 grid used in Section 3.2.2 (see Fig. C.1(b)).

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Domain Triangulation: 20 × 20

y

(a) 20× 20 grid (showing elements)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Domain Triangulation:MxN=30×30

y

(b) 30× 30 grid (showing elements)

Figure C.1: The two finite element grids (2D) that were used

63

C.1 Varying mass

C.1.1 Computational effort/cost

Fig. C.2 depicts the work (as measured by the number of iterations of the fixed

point method) that was required to resolve the non-linear portion of the problem

to the tolerance (TOL) specified in Table 3.1 for the two mass-variance experiments.

The low iteration counts at late times in Fig. C.2 suggest that the solutions at the

ends of these time-steps served as excellent initial guesses for the fixed point method

at the start of each subsequent step. Given the relatively large time-step used for

these experiments, we were able to simulate relatively long time periods in these

experiments (∼ 12 seconds) and, evidently, achieve quite stable end-state solutions.

0 2 4 6 8 10 120

5

10

15

20

25

30

35

time

# Ite

ratio

ns

Convergence iterations #elements=20

(a) m = 0

0 2 4 6 8 10 120

5

10

15

20

25

30

time

# Ite

ratio

ns

Convergence iterations #elements=20

(b) m = 0.4

Figure C.2: Non-linear work (2D) for various m

The contrast with most of the results depicted in Fig. C.5, in which we simulate

approximately one second in each experiment is quite stark and suggests that the

end-state solutions returned for these latter experiments could well be metastable.

64

C.1.2 Component energy evolution

Fig. C.3 depicts the per-component free energy evolution that resulted from the ex-

periments conducted in Section 3.2.1. The symmetric case (m = 0) is illustrated in

Fig. C.3(a) and the non-symmetric case (m 6= 0) in Fig. C.3(b).

0 5 100

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 5 100

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Part 2

0 5 100

0.01

0.02

0.03

0.04

time

Fre

e E

nerg

y

Part 3

(a) m = 0

0 5 100

0.05

0.1

time

Fre

e E

nerg

y

Part 1

0 5 100

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Part 2

0 5 100

0.005

0.01

0.015

0.02

time

Fre

e E

nerg

yPart 3

(b) m = 0.4

Figure C.3: Typical component energy evolution over time (2D)

65

C.2 Varying the non-local energy coefficient

C.2.1 The initial condition

The initial condition that was used for the four experiments discussed in Section 3.2.2

can be seen in Fig. C.4 (it was taken from the test result set returned for the σ = 800

test). The mass asymmetry (we experiment with m = 0.4) is visually clear in the

‘colorbar’ at the bottom of the figure.

x

y

ε=0.02, σ=800 and m=0.4 at t=0

0 0.5 10

0.2

0.4

0.6

0.8

1

−0.5 0 0.5 1

Figure C.4: Initial condition used to test the effect of varying σ

C.2.2 Computational effort/cost

Computational cost/effort graphs summarising the four tests of Section 3.2.2 can be

seen in Fig. C.5. Fig. C.5(d) relates to the disordered case and resembles the results

depicted in Fig. C.2 suggesting that this test ended at a highly stable end-state. The

results in the other three graphs suggest a distinct lack of stability, however, as the

66

non-linear fixed point algorithm is still ‘working very hard’ (hundreds of iterations)

to resolve each time step. These results are not unexpected if we compare the to-

tal simulated time in the experiments of Appendix C.1.1 (∼ 13 seconds) with those

here (approximately one second each). They are somewhat difficult, intuitively, to

reconcile with the results depicted in the next section in which we see free energy ap-

parently hardly changing. These phenomena have been reported by other researchers

as well – see for instance Fig. 12 of [CPW09].

0 0.2 0.4 0.6 0.8 1 1.20

50

100

150

200

250

300

time

# Ite

ratio

ns

Convergence iterations #elements=30

(a) σ = 2

0 0.2 0.4 0.6 0.80

20

40

60

80

100

time

# Ite

ratio

ns

Convergence iterations #elements=30

(b) σ = 20

0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

35

40

45

time

# Ite

ratio

ns

Convergence iterations #elements=30

(c) σ = 200

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

time

# Ite

ratio

ns

Convergence iterations #elements=30

(d) σ = 800 (‘disorder’)

Figure C.5: Non-linear work (2D) for various σ

67

C.2.3 Total energy evolution

Fig. C.6 below depicts the total free energy evolution for each of the four experiments

discussed in Section 3.2.2. We notice immediately that each of these functions is

a non-increasing function of time – as expected. In addition, in all four cases, the

energy initially drops quickly and then remains quite stable. An exception is noted

in Fig. C.6(a) which we explore further in Appendix C.2.4 below.

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(a) σ = 2

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(b) σ = 20

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(c) σ = 200

0 0.01 0.02 0.03 0.04 0.050

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(d) σ = 800 (‘disorder’)

Figure C.6: Total free energy evolution (2D) for various σ

C.2.4 The extended run for σ = 2

We include here summary results from extending the experiment conducted for σ = 2

in Section 3.2.2, for an additional 2 000 time-steps (equivalent to a further 0.8 sim-

68

ulated seconds). The end-state solution is depicted in Fig. C.7(a) and the total free

energy evolution in Fig. C.7(b). Evidently very little changes during this additional

time. Given that it took ∼ 60 hours to obtain these results, we defer further exam-

ination of this experiment to a significantly more powerful computer than the one

used thus far in these experiments.

x

yε=0.02, σ=2 and m=0.4 at t=0.8

0 0.5 10

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5(a) End-state solution u(x, 2); the graph title

shows t = 0.8 because the simulation was re-

started at t = 0 with the output from the origi-

nal experiment

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

time

Fre

e E

nerg

y

Total Free Energy Evolution over Time

(b) Total free energy evolution for σ = 2 from

t = 1.2 secs to t = 2.0 secs

Figure C.7: Extended run results for σ = 2 (2D)

69

C.2.5 h-independence

We conclude with a basic investigation of h-independence to see if our results are in

some way an artifact of the underlying discretisation of Ω. In the short term, this

investigation will serve as a proxy for a formal analysis of the error and convergence

properties of our numerical method.

As noted previously, our finite element code is quite performance intensive and

so working on finer meshes than those depicted in Fig. C.1 is difficult. Nevertheless,

we interpolate1 the initial condition from the σ = 200 experiment from Section 3.2.2

onto a 100 × 100 mesh and then re-execute the experiment for 500 time-steps. We

compare the results on the fine grid to those obtained on the 30× 30 mesh after the

500th time-step in Fig. C.8. A few observations follow:

• It took ∼ 16 hours to obtain the results depicted in Fig. C.8(b) and so further

analysis over longer simulated time horizons is deferred until we have access to

a more powerful computer.

• The contour lines on the 100 × 100 mesh are smoother than their 30 × 30

counterparts – as expected.

• We notice that the coarse 30× 30 mesh is not a subset of the (finer) 100× 100

mesh. Consequently, we retain the same average mass in going to the finer mesh,

but the interpolation has the effect of smoothing the initial condition some-

what.2 We see the effect of this in the energy evolution diagrams in Fig. C.8,

where the initial free energy is somewhat lower on the 100×100 mesh. A detailed

investigation reveals that the ‘Part 1’ energy (related to the gradient on each

element) is lower on the fine mesh, whereas the initial ‘Part 2’ and ‘Part 3’ en-

ergies are largely mesh-independent. The pattern of energy evolution, however,

is quite similar once the numerical method takes over.

• Each of the physical ‘spots’ present in Fig. C.8(a) has a corresponding structure

in Fig. C.8(b) in approximately the same location in Ω. The two spots in the top

and bottom left corners (see the red arrows on the two sub-figures in Fig. C.8)

are in slightly different positions, but there is good empirical evidence to suggest

1We use the ‘interp2’ function in Matlab for this.2Subsequent investigation also revealed that the Matlab ‘interp2’ function does not interpolate

onto finer grids in the way we would want it to for a triangulation (in estimating values at new

nodes, it averages in all directions).

70

that these two solutions are heading towards essentially the same (stable) end-

state.

The same test was executed for the σ = 800 ‘disorder’ case from Fig. 3.5(d). The 100×100 mesh results match those observed on the 30× 30 mesh very closely, suggesting

that our results are not seriously affected by the discretisation on which we work

(notwithstanding our approach to interpolation onto the finer grid with ‘interp2’).

x

y

ε=0.02, σ=200 and m=0.4 at t=0.2000

→

→

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time

Fre

e E

nerg

y

Free Energy Evolution Over Time

−0.5 0 0.5

(a) 30× 30 grid

x

y

ε=0.02, σ=200 and m=0.4 at t=0.2000

→

→

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

0.25

0.3

time

Fre

e E

nerg

y

Free Energy Evolution Over Time

−0.5 0 0.5

(b) 100× 100 grid

Figure C.8: Results for σ = 200 (2D) after 500 time steps on two different meshes

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