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1 Title of your thesis here Your name here A Thesis presented for the degree of Doctor of Philosophy Your Research Group Here Department of Mathematical Sciences University of Durham England Month and Year

2 Dedicated to Someone here

3 Title of your thesis here Your name here Submitted for the degree of Doctor of Philosophy month year here Abstract Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here Your abstract here

4 Declaration The work in this thesis is based on research carried out at the YOUR RESEARCH GROUP HERE, the Department of Mathematical Sciences, England. No part of this thesis has been submitted elsewhere for any other degree or qualification and it is all my own work unless referenced to the contrary in the text. [If your thesis based on joint research, you have to mention what part of it is your individual constribution, see Rules for the Submission of Work for Higher Degrees for detail. You will get one from the Graduate School.] Copyright c 2001 by YOUR NAME HERE. The copyright of this thesis rests with the author. No quotations from it should be published without the author s prior written consent and information derived from it should be acknowledged. iv

5 Acknowledgements Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template.thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. Thank to someone who prepared this template. v

6 Contents Abstract Declaration Acknowledgements iii iv v 1 Introduction 1 2 Evolutionary problem Notation The Existence and Uniqueness of the Continuous Problem A Semidiscrete approximation Notations A fully discrete approximation Scheme Existence and Uniqueness Numerical Experiments Practical Algorithms Iterative Method for Scheme Conclusions 11 Bibliography 12 Appendix 13 vi

7 Contents vii A Basic and Auxiliary Results 13 A.1 Basic Results October 10, 2014

8 List of Figures viii

9 List of Tables ix

10 Chapter 1 Introduction Let Ω be bounded domain in R d (d 3) with Lipschitz boundary Ω. We consider a coupled pair of Cahn-Hilliard Equations modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component denoted by A and the other by B: Find {u 1 (x, t), u 2 (x, t)} R R such that where u 1 t = w 1 in Ω, t > 0, (1.0.1a)[1a0001a] u 2 t = w 2 in Ω, t > 0, (1.0.1b)[1a0001b] w 1 = δf (u 1, u 2 ), δu 1 (1.0.1c)[1a0001c] w 2 = δf (u 1, u 2 ), δu 2 (1.0.1d)[1a0001d] F (u 1, u 2 ) = b 1 u 4 1 a 1 u c 1 u b 2 u 4 2 a 2 u c 2 u 2 2 ( ) 2 ( a1 + D u 1 + u 2 + 2b 1 a2 2b 2 ) 2.(1.0.1e)[1a0001e] Here δf (u 1, u 2 )/δu i, for i = 1, 2, indicates the functional derivative. The variable u 1 denotes a local concentration of A or B and u 2 indicates the presence of a liquid or a vapour phase. c i denote the surface tension of u i. The coefficient a i is proportional to T ci T, where T c1 corresponds to the critical temperature of the A-B phase 1

11 Chapter 1. Introduction 2 separation, and T c2 separation. represents the critical temperature of the liquid-vapour phase If a 1 > 0, a 2 > 0, there are two equilibrium phases for each field corresponding to u 1 = ± a1 2b 1 and u 2 = ± a2 2b 2, denote as u + 1, u 1, u + 2, and u 2. The coupling D energetically inhibits the existence of the phase denoted by the (u + 1, u + 2 ). Thus we have three-phase system: liquid A correspond to (u 1, u 2 ) regions, liquid B to (u + 1, u 2 ) regions and the vapour phase to (u 1, u + 2 ) regions. If D = 0, the problem reduces to two decoupled Cahn-Hilliard equations, which has been discussed at length in the mathematical literature. And for this type of problem, we do not have liquid-vapour interfaces. In Chapter 5 two practical algorithms (implicit an explicit methods) for solving the finite element at each time step are suggested. We discuss the convergence theory for the implicit scheme, which used to solve the system arising from Scheme 1. We also discuss in this chapter some computational results for one and two space dimensions. We only use the implicit scheme for all simulations. Before showing some computational results, we discuss linear stability solutions in one space dimension. October 10, 2014

12 Chapter 2 Evolutionary problem In this chapter a global existence and uniqueness theorem for a weak formulation possessing a Lyapunov function is proven. Regularity results are presented for the weak solution. 2.1 Notation Let Ω be a bounded domain in R d, d 3 with boundary Ω. For d = 2, 3 we assume that Ω is Lipschitz boundary. Through out this thesis we adopt the standard notation for Sobolev spaces, denoting the norm of W m,p (Ω) (m N, p [1, ]) by m,p and semi-norm by m,p. For p = 2, W m,p (Ω) will be denoted by H m with the associated norm and semi-norm written as m and m, respectively. In addition we denote the L 2 (Ω) inner product over Ω by (, ) and define the mean of integral η := 1 (η, 1) Ω η L1 (Ω). We also use the following notation, for 1 q <, { L q (0, T ; W m,p (Ω)) := η(x, t) : η(, t) W m,p (Ω), { L (0, T ; W m,p (Ω)) := T 0 } η(, t) q m,p dt <, } η(x, t) : η(, t) W m,p (Ω), ess sup η(, t) m,p < t (0,T ) For later purposes, we recall the Hölder inequality for u L p, v L q 1 < p <, ( uv dx Ω u p dx Ω ) 1 p ( Ω ) 1 v q q 1 dx, where 3 p + 1 q = 1,, and (2.1.1)[Holder]

13 2.2. The Existence and Uniqueness of the Continuous Problem 4 and the following well-known Sobolev interpolation results: Let p [1, ], m 1 ( ) and v W m,p (Ω). Then there are constants C and µ = d 1 1 such that the m p r inequality v 0,r C v 1 µ 0,p v µ m,p, holds for r [p, ] if m d > 0, p [p, ) if m d = 0, p [p, d ] if m d < 0. m d/p p (2.1.2)[gagliar] We also state the following lemma, which will prove useful in our subsequent analysis. Lemma [Lem201] Let u, v, η H 1 (Ω), f = u v, g = u m v n m, m, n = 0, 1, 2, and n m 0. Then for d = 1, 2, 3, fgηdx C u v 0 u m 1 Ω v n m 1 η 1. (2.1.3)[2le000] Proof: Note that using the Cauchy-Schwarz inequality we have u m (u) m v n m 0,2mp v (n m) 0,2(n m)p for n m 0, and m 0, 0,p u m 0,mp or v (n m) 0,(n m)p for m = 0, or n m = 0 respectively. Noting the generalise Hölder inequality and the result above we have fgηdx u v 0 u m v n m 0,3 η 0,6, Ω u 0,6 for m = 2, u v 0 η 0,6 u 0,6 v 0,6 for m = 1, v 2 0,6 for m = 0, C u v 0 u m 1 v n m 1 η 1, where we have noted (2.1.2) to obtain the last inequality. This ends the proof. 2.2 The Existence and Uniqueness of the Continuous Problem Given γ > 0 and u 0 i H 1 (Ω), for i = 1, 2, such that u u C. We consider the problem: October 10, 2014

14 2.2. The Existence and Uniqueness of the Continuous Problem 5 (P) Find {u i, w i } such that u i H 1 (0, T ; (H 1 (Ω)) ) L (0, T ; H 1 (Ω)) for a.e. t (0, T ), w i L 2 (0, T ; H 1 (Ω)) u1 t, η = ( w 1, η), (2.2.1a)[2B0000a] (w 1, η) = (φ(u 1 ), η)+ γ( u 1, η) + 2D(Ψ 1 (u 1, u 2 ), η),(2.2.1b)[2b0000b] u 1 (x, 0) = u 0 1(x), (2.2.1c)[2B0000c] and u2 t, η = ( w 2, η), (2.2.1d)[2B0000d] (w 2, η) = (φ(u 2 ), η)+ γ( u 2, η) + 2D(Ψ 2 (u 1, u 2 ), η),(2.2.1e)[2b0000e] u 2 (x, 0) = u 0 2(x), (2.2.1f)[2B0000f] for all η H 1 (Ω) for a.e. t (0, T ). October 10, 2014

15 Chapter 3 A Semidiscrete approximation In this chapter we introduce some notation which will be used in the current and following chapters. For completeness, we prove interpolation error estimates in the finite element space as these are necessary tools for analysis the current chapter and chapter 4. Then a semidiscrete finite element approximation is proposed where the existence is shown and the uniqueness is proven for one and two dimensions. An error bound between the semidiscrete and continuous solution is given is the final section. 3.1 Notations Let S h H 1 (Ω) be finite element space defined by S h := {χ C( Ω) : χ τ is linear τ T h } H 1 (Ω). Denote by {x i } J i=1 the set of nodes of T h and let {η i } J i=1 be basis for S h defined by η i (x j ) = δ ij, for i, j = 1,..., J. Let π h : C( Ω) S h be the interpolation operator such that π h χ(x i ) = χ(x i ), for i = 1,..., J and define a discrete inner product on C( Ω) as follows J (χ 1, χ 2 ) h := π h (χ 1 (x)χ 2 (x))dx m i χ 1 (x i )χ 2 (x i ), (3.1.1)[3S0000] Ω i=1 where m i = (η i, η i ) h. The induced norm h := [(, ) h ] 1 2 on S h is equivalent to 0 := [(, )] 1 2. Note that the integral (3.1.1) can easily be computed by means of vertex quadrature rule, which exact for piecewise linear functions. 6

16 Chapter 4 A fully discrete approximation In this chapter we introduce a numerical scheme (we call scheme 1) to solve the weak formulation form we mentioned in Chapter 1. We discussed the existence and uniqueness of the solutions for the scheme. We also discuss stability and convergence of the solution to the continuous problem in the weak formulation. We briefly mention a second scheme (we call scheme 2) and show existence, uniqueness, and the stability. We do not discuss the convergence of the scheme. 4.1 Scheme Existence and Uniqueness Given N, a positive integer, let t = T/N denote a fixed time step, and t k = k t where k = 0,..., N. We focus our attention on approximating (P) by the discrete scheme defined as follows: (P h, t 1 ) Given U 0 1, U 0 2, find {U n 1, U n 2, W n 1, W n 2 } S h S h S h S h, for n = 1,..., N, such that η S h ( U 1 n U1 n 1, η) h = ( W1 n, η), (4.1.1a)[4F0008] t (W n 1, η) h = (F 1 (U n 1, U n 2 ),η) h + γ( U n 1, η), (4.1.1b)[4F0009] U 0 1 = P h u 0 1, (4.1.1c)[4F0010] 7

17 4.1. Scheme 1 8 and ( U 2 n U2 n 1, η) h = ( W2 n, η), (4.1.1d)[4F0011] t (W n 2, η) h = (F 2 (U n 1, U n 2 ),η) h + γ( U n 2, η), (4.1.1e)[4F0012] U 0 2 = P h u 0 2, (4.1.1f)[4F0013] where F 1 (U1 n, U2 n ) = (U1 n ) 3 U1 n 1 + D(U1 n + U1 n 1 + 2)(U2 n 1 + 1) (4.1.1g)[4F0014] 2, F 2 (U n 1, U n 2 ) = (U n 2 ) 3 U n D(U n 2 + U n )(U n 1 + 1) 2.(4.1.1h)[4F0015] Note that (4.1.1a c) is independent of U n 2 and (4.1.1d f) is dependent on U n 1. October 10, 2014

18 Chapter 5 Numerical Experiments In this chapter we discuss two practical algorithms (implicit and explicit method) that are used to solve an algebraic system arising in the problem discussed in this thesis. We discuss the convergence theory for the implicit scheme as used to solve the system arising from the scheme 1. We also discuss some computational results for one and two dimensions. We use the implicit scheme in all simulations in this chapter. We have made a comparison with Scheme 2 and the results are compatible. Before showing some computational results, we discuss linear stability solution for the problem. 5.1 Practical Algorithms Iterative Method for Scheme 1 [implicit] Let us expand U i and W i, i = 1, 2, in terms of the standard nodal basis functions of the finite element space S h, that is, U n 1 = U n 2 = J U1,iη n i, W1 n = i=1 J U2,iη n i, W2 n = i=1 J W1,iη n i, i=1 J W2,iη n i, i=1 (5.1.1a)[5E0001a] (5.1.1b)[5E0001b] where J be the number of node points. 9

19 5.1. Practical Algorithms 10 Concluding Remarks To see a clear interaction between the solutions U 1 and U 2 in terms of their physical meaning, it is worthwhile doing computational experiment in three space dimensions. October 10, 2014

20 Chapter 6 Conclusions It was shown using a Faedo Galerkin approximation that there exist a unique solution for the coupled pair of Cahn-Hilliard equations modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. This solution satisfies certain stability bounds. The regularity result was driven at the end of Chapter 2 as having major contribution in obtaining the error bound for the method proposed. 11

21 Bibliography [1] J. W. Barrett and J. F. Blowey (1995), An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numerische Mathematics, 72, pp [2] J. W. Barrett and J. F. Blowey (1997), Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy, Numerische Mathematics, 77, pp [3] J. W. Barrett and J. F. Blowey (1999a), An improve error bound for finite element approximation of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal. 19, pp [4] P. G. Ciarlet (1978), The Finite Element Method for Elliptic Problems, North-Holland. [5] J. L. Lions (1969), Quelques Móthodes de Résolution des Problémes aux Limites, Dunod. 12

22 Appendix A Basic and Auxiliary Results A.1 Basic Results 13