Finite Element Methods for Non-Fickian Diffusion in Viscoelastic...

Finite Element Methods for Non-FickianDiffusion in Viscoelastic Polymers

A thesis submitted for the degree ofDoctor of Philosophy

byNorbert Bauermeister

School of Information Systems, Computing and MathematicsBrunel University

December 2007

Brunel University, Uxbridge Norbert BauermeisterSchool of Information Systems, PhD thesis

Computing and Mathematics December 2007

Finite Element Methods for Non-FickianDiffusion in Viscoelastic Polymers

Abstract

The nonlinear problem of non-Fickian polymer diffusion is considered. The model istaken from Cohen, White & Witelski, SIAM J. Appl. Math. 55, pp. 348–368, 1995. Itconsists of a diffusion equation with an adjoined spatially local evolution equation fora viscoelastic stress. In their simplest form the equations are

ut −∇2u−∇2σ = 0,σt + γ(u)σ = u,

for certain bounded functions γ : R → R.Numerical schemes are presented that are based, spatially, on the Galerkin finite

element method and, temporally, on the Crank-Nicolson method. Special attentionis paid to linearising the discrete equations by extrapolating the value of the discretesolution from previous timesteps. Optimal a priori error estimates are given, basedon the assumption that the exact solution and the domain possess certain regularityproperties. The error estimate is, in the case of approximation by piecewise linearpolynomials,

maxn

(

‖u(tn) − uhn‖0 + ‖σ(tn) − σhn‖0)

+h(

∑

n

‖ūn + σ̄n − ūhn − σ̄hn‖21)1/2

≤ C(k2 + h2),

where v̄n =12(v(tn) + v(tn−1)) for v ∈ {u, σ, uh, σh}. Numerical experiments are given

to support the error estimates.For the variational formulation that forms the basis of the discrete schemes the

existence and uniqueness of solutions in one space dimension is established.Further numerical experiments are included to show some of the behaviour of the

solutions of the model.

Contents

List of Tables 6

List of Figures 9

List of Theorems 11

Acknowledgements 13

1 Introduction 141.1 Overview of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Mathematical Background 162.1 Elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 The domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Sobolev spaces and their embeddings . . . . . . . . . . . . . . . . 172.1.3 Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 Inverse Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Time dependent problems . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Spaces for time dependent problems . . . . . . . . . . . . . . . . 232.2.2 Unique solution of ODEs . . . . . . . . . . . . . . . . . . . . . . 252.2.3 Linear parabolic equations . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Space discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.1 Mesh and finite element spaces . . . . . . . . . . . . . . . . . . . 262.3.2 Inverse estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 L2-projection and elliptic projection . . . . . . . . . . . . . . . . 30

3 The Cohen Model for Non-Fickian Diffusion 333.1 The Cohen model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Physical interpretation of the model . . . . . . . . . . . . . . . . . . . . 343.3 Notes on literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Rescaling the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 The nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.1 Choices for nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 373.5.2 Basic properties of γ . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Existence and Uniqueness 424.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1 Two approaches for getting results . . . . . . . . . . . . . . . . . 434.2 Boundedness of exact solutions . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Using the inverse Laplacian . . . . . . . . . . . . . . . . . . . . . 44

3

CONTENTS 4

4.2.2 Using the u+ σ-approach . . . . . . . . . . . . . . . . . . . . . . 454.3 Unique solution of linearised problem . . . . . . . . . . . . . . . . . . . . 48

4.3.1 Weighted norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Unique solution in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.1 An iteration for the nonlinear problem . . . . . . . . . . . . . . . 524.4.2 The fixed point of the iteration . . . . . . . . . . . . . . . . . . . 564.4.3 What about d = 2 and d = 3? . . . . . . . . . . . . . . . . . . . . 58

5 Discretisation and Numerical Scheme 595.1 Time discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1.2 Taylor estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.3 Discrete inverse Laplacian . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3.1 Matrix form for Q = 3 . . . . . . . . . . . . . . . . . . . . . . . . 665.4 About uniqueness of discrete solutions . . . . . . . . . . . . . . . . . . . 67

5.4.1 Methods Q ∈ {1, 2} . . . . . . . . . . . . . . . . . . . . . . . . . 675.4.2 Method Q = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.3 Methods Q ∈ {4, 5} . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 Boundedness and stability of discrete solution . . . . . . . . . . . . . . . 715.5.1 Bound on solution . . . . . . . . . . . . . . . . . . . . . . . . . . 715.5.2 Bound on differences . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Error Estimate 796.1 Preparations for the error estimate . . . . . . . . . . . . . . . . . . . . . 796.2 Error inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3 Nonlinearity errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4.1 Why so complicated? . . . . . . . . . . . . . . . . . . . . . . . . 986.5 About method Q = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Numerical Results 1017.1 Another discrete formulation . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2.1 Defining errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.3 Testing the programs for exactness . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 Degree 4 polynomials in 3D . . . . . . . . . . . . . . . . . . . . . 1047.4 Convergence tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.4.1 Overview of tables . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.4.2 Influence of time step k for Q ∈ {4, 5} . . . . . . . . . . . . . . . 1067.4.3 Summary of the tables . . . . . . . . . . . . . . . . . . . . . . . . 106

7.5 Behaviour of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5.1 Sharp front in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5.2 Infinite domain in 1D: Ω = (0,∞) . . . . . . . . . . . . . . . . . 1447.5.3 Wave-like solution in 1D . . . . . . . . . . . . . . . . . . . . . . . 1587.5.4 Sharp front in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.5.5 Ring-shaped domain in 2D . . . . . . . . . . . . . . . . . . . . . 1677.5.6 Sharp front in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

CONTENTS 5

8 Conclusion 1738.1 Suggestions for further work . . . . . . . . . . . . . . . . . . . . . . . . . 173

A More on Mathematical Background 175A.1 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175A.2 Banach’s fixed point theorem . . . . . . . . . . . . . . . . . . . . . . . . 176A.3 Bounds for norms in finite-dimensional spaces . . . . . . . . . . . . . . . 177A.4 Gelfand triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180A.5 Positive definite matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 183

B Amann’s Result of Existence and Uniqueness 185

References 188

List of Symbols 191

Index 196

List of Tables

7.1 k-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1: ecomb . . . . . . . . 1087.2 k-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1: e0, e1 . . . . . . . . 1087.3 k-convergence for σD = 1, γ3, Q = 2, d = 2, r = 1: ecomb . . . . . . . . 1087.4 k-convergence for σD = 1, γ3, Q = 2, d = 2, r = 1: e0, e1 . . . . . . . . 1097.5 k-convergence for σD = 1, γ3, Q = 3, d = 2, r = 1: ecomb . . . . . . . . 1097.6 k-convergence for σD = 1, γ3, Q = 3, d = 2, r = 1: e0, e1 . . . . . . . . 1097.7 k-convergence for σD = 1, γ3, Q = 4, d = 2, r = 1: ecomb . . . . . . . . 1107.8 k-convergence for σD = 1, γ3, Q = 4, d = 2, r = 1: e0, e1 . . . . . . . . 1107.9 k-convergence for σD = 1, γ3, Q = 5, d = 2, r = 1: ecomb . . . . . . . . 1107.10 k-convergence for σD = 1, γ3, Q = 5, d = 2, r = 1: e0, e1 . . . . . . . . 1117.11 k-convergence for σD = 1, γ1, Q = 1, d = 2, r = 1: ecomb . . . . . . . . 1117.12 k-convergence for σD = 1, γ1, Q = 1, d = 2, r = 1: e0, e1 . . . . . . . . 1117.13 k-convergence for σD = 1, γ1, Q = 2, d = 2, r = 1: ecomb . . . . . . . . 1127.14 k-convergence for σD = 1, γ1, Q = 2, d = 2, r = 1: e0, e1 . . . . . . . . 1127.15 k-convergence for σD = 1, γ1, Q = 3, d = 2, r = 1: ecomb . . . . . . . . 1127.16 k-convergence for σD = 1, γ1, Q = 3, d = 2, r = 1: e0, e1 . . . . . . . . 1137.17 k-convergence for σD = 1, γ1, Q = 4, d = 2, r = 1: ecomb . . . . . . . . 1137.18 k-convergence for σD = 1, γ1, Q = 4, d = 2, r = 1: e0, e1 . . . . . . . . 1137.19 k-convergence for σD = 1, γ1, Q = 5, d = 2, r = 1: ecomb . . . . . . . . 1147.20 k-convergence for σD = 1, γ1, Q = 5, d = 2, r = 1: e0, e1 . . . . . . . . 1147.21 k-convergence for σD = 0, γ3, Q = 1, d = 2, r = 1: ecomb . . . . . . . . 1147.22 k-convergence for σD = 0, γ3, Q = 1, d = 2, r = 1: e0, e1 . . . . . . . . 1157.23 k-convergence for σD = 0, γ3, Q = 2, d = 2, r = 1: ecomb . . . . . . . . 1157.24 k-convergence for σD = 0, γ3, Q = 2, d = 2, r = 1: e0, e1 . . . . . . . . 1157.25 k-convergence for σD = 0, γ3, Q = 3, d = 2, r = 1: ecomb . . . . . . . . 1167.26 k-convergence for σD = 0, γ3, Q = 3, d = 2, r = 1: e0, e1 . . . . . . . . 1167.27 k-convergence for σD = 0, γ3, Q = 4, d = 2, r = 1: ecomb . . . . . . . . 1167.28 k-convergence for σD = 0, γ3, Q = 4, d = 2, r = 1: e0, e1 . . . . . . . . 1177.29 k-convergence for σD = 0, γ3, Q = 5, d = 2, r = 1: ecomb . . . . . . . . 1177.30 k-convergence for σD = 0, γ3, Q = 5, d = 2, r = 1: e0, e1 . . . . . . . . 1177.31 k-convergence for σD = 0, γ1, Q = 1, d = 2, r = 1: ecomb . . . . . . . . 1187.32 k-convergence for σD = 0, γ1, Q = 1, d = 2, r = 1: e0, e1 . . . . . . . . 1187.33 k-convergence for σD = 0, γ1, Q = 2, d = 2, r = 1: ecomb . . . . . . . . 1187.34 k-convergence for σD = 0, γ1, Q = 2, d = 2, r = 1: e0, e1 . . . . . . . . 1197.35 k-convergence for σD = 0, γ1, Q = 3, d = 2, r = 1: ecomb . . . . . . . . 1197.36 k-convergence for σD = 0, γ1, Q = 3, d = 2, r = 1: e0, e1 . . . . . . . . 1197.37 k-convergence for σD = 0, γ1, Q = 4, d = 2, r = 1: ecomb . . . . . . . . 1207.38 k-convergence for σD = 0, γ1, Q = 4, d = 2, r = 1: e0, e1 . . . . . . . . 1207.39 k-convergence for σD = 0, γ1, Q = 5, d = 2, r = 1: ecomb . . . . . . . . 1207.40 k-convergence for σD = 0, γ1, Q = 5, d = 2, r = 1: e0, e1 . . . . . . . . 121

6

LIST OF TABLES 7

7.41 k-convergence for σD = 1, γ3, Q = 1, d = 1, r = 1: ecomb . . . . . . . . 1217.42 k-convergence for σD = 1, γ3, Q = 1, d = 1, r = 1: e0, e1 . . . . . . . . 1217.43 k-convergence for σD = 1, γ3, Q = 1, d = 3, r = 1: ecomb . . . . . . . . 1227.44 k-convergence for σD = 1, γ3, Q = 1, d = 3, r = 1: e0, e1 . . . . . . . . 1227.45 h-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1, k = 1.0: ecomb . . . 1227.46 h-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1, k = 1.0: e0, e1 . . . 1237.47 h-convergence for σD = 1, γ3, Q = 2, d = 2, r = 1, k = 1.0: ecomb . . . 1237.48 h-convergence for σD = 1, γ3, Q = 2, d = 2, r = 1, k = 1.0: e0, e1 . . . 1237.49 h-convergence for σD = 1, γ3, Q = 3, d = 2, r = 1, k = 1.0: ecomb . . . 1247.50 h-convergence for σD = 1, γ3, Q = 3, d = 2, r = 1, k = 1.0: e0, e1 . . . 1247.51 h-convergence for σD = 1, γ3, Q = 4, d = 2, r = 1, k = 0.0001: ecomb . . 1247.52 h-convergence for σD = 1, γ3, Q = 4, d = 2, r = 1, k = 0.0001: e0, e1 . 1257.53 h-convergence for σD = 1, γ3, Q = 5, d = 2, r = 1, k = 0.0001: ecomb . . 1257.54 h-convergence for σD = 1, γ3, Q = 5, d = 2, r = 1, k = 0.0001: e0, e1 . 1257.55 h-convergence for σD = 1, γ1, Q = 1, d = 2, r = 1, k = 1.0: ecomb . . . 1267.56 h-convergence for σD = 1, γ1, Q = 1, d = 2, r = 1, k = 1.0: e0, e1 . . . 1267.57 h-convergence for σD = 1, γ1, Q = 2, d = 2, r = 1, k = 1.0: ecomb . . . 1267.58 h-convergence for σD = 1, γ1, Q = 2, d = 2, r = 1, k = 1.0: e0, e1 . . . 1277.59 h-convergence for σD = 1, γ1, Q = 3, d = 2, r = 1, k = 1.0: ecomb . . . 1277.60 h-convergence for σD = 1, γ1, Q = 3, d = 2, r = 1, k = 1.0: e0, e1 . . . 1277.61 h-convergence for σD = 1, γ1, Q = 4, d = 2, r = 1, k = 0.0001: ecomb . . 1287.62 h-convergence for σD = 1, γ1, Q = 4, d = 2, r = 1, k = 0.0001: e0, e1 . 1287.63 h-convergence for σD = 1, γ1, Q = 5, d = 2, r = 1, k = 0.0001: ecomb . . 1287.64 h-convergence for σD = 1, γ1, Q = 5, d = 2, r = 1, k = 0.0001: e0, e1 . 1287.65 h-convergence for σD = 0, γ3, Q = 1, d = 2, r = 1, k = 1.0: ecomb . . . 1297.66 h-convergence for σD = 0, γ3, Q = 1, d = 2, r = 1, k = 1.0: e0, e1 . . . 1297.67 h-convergence for σD = 0, γ3, Q = 2, d = 2, r = 1, k = 1.0: ecomb . . . 1297.68 h-convergence for σD = 0, γ3, Q = 2, d = 2, r = 1, k = 1.0: e0, e1 . . . 1307.69 h-convergence for σD = 0, γ3, Q = 3, d = 2, r = 1, k = 1.0: ecomb . . . 1307.70 h-convergence for σD = 0, γ3, Q = 3, d = 2, r = 1, k = 1.0: e0, e1 . . . 1307.71 h-convergence for σD = 0, γ3, Q = 4, d = 2, r = 1, k = 0.0001: ecomb . . 1317.72 h-convergence for σD = 0, γ3, Q = 4, d = 2, r = 1, k = 0.0001: e0, e1 . 1317.73 h-convergence for σD = 0, γ3, Q = 5, d = 2, r = 1, k = 0.0001: ecomb . . 1327.74 h-convergence for σD = 0, γ3, Q = 5, d = 2, r = 1, k = 0.0001: e0, e1 . 1327.75 h-convergence for σD = 0, γ1, Q = 1, d = 2, r = 1, k = 1.0: ecomb . . . 1327.76 h-convergence for σD = 0, γ1, Q = 1, d = 2, r = 1, k = 1.0: e0, e1 . . . 1337.77 h-convergence for σD = 0, γ1, Q = 2, d = 2, r = 1, k = 1.0: ecomb . . . 1337.78 h-convergence for σD = 0, γ1, Q = 2, d = 2, r = 1, k = 1.0: e0, e1 . . . 1337.79 h-convergence for σD = 0, γ1, Q = 3, d = 2, r = 1, k = 1.0: ecomb . . . 1347.80 h-convergence for σD = 0, γ1, Q = 3, d = 2, r = 1, k = 1.0: e0, e1 . . . 1347.81 h-convergence for σD = 0, γ1, Q = 4, d = 2, r = 1, k = 0.0001: ecomb . . 1347.82 h-convergence for σD = 0, γ1, Q = 4, d = 2, r = 1, k = 0.0001: e0, e1 . 1357.83 h-convergence for σD = 0, γ1, Q = 5, d = 2, r = 1, k = 0.0001: ecomb . . 1357.84 h-convergence for σD = 0, γ1, Q = 5, d = 2, r = 1, k = 0.0001: e0, e1 . 1357.85 h-convergence for σD = 1, γ3, Q = 1, d = 1, r = 1, k = 1.0: ecomb . . . 1367.86 h-convergence for σD = 1, γ3, Q = 1, d = 1, r = 1, k = 1.0: e0, e1 . . . 1367.87 h-convergence for σD = 1, γ3, Q = 1, d = 3, r = 1, k = 1.0: ecomb . . . 1377.88 h-convergence for σD = 1, γ3, Q = 1, d = 3, r = 1, k = 1.0: e0, e1 . . . 1377.89 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 2, k = 1.0: ecomb . . . 1377.90 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 2, k = 1.0: e0, e1 . . . 1387.91 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 3, k = 1.0: ecomb . . . 139

LIST OF TABLES 8

7.92 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 3, k = 1.0: e0, e1 . . . 1397.93 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 4, k = 1.0: ecomb . . . 1407.94 h-convergence for σD = 1, γ3, Q = 3, d = 1, r = 4, k = 1.0: e0, e1 . . . 1407.95 h-convergence for σD = 1, γ3, Q = 4, d = 1, r = 1, k = 0.0001: ecomb . . 1407.96 h-convergence for σD = 1, γ3, Q = 4, d = 1, r = 1, k = 0.0001: e0, e1 . 1417.97 h-convergence for σD = 1, γ3, Q = 4, d = 1, r = 1, k = 0.25: ecomb . . . 1417.98 h-convergence for σD = 1, γ3, Q = 4, d = 1, r = 1, k = 0.25: e0, e1 . . . 1417.99 h-convergence for σD = 1, γ3, Q = 5, d = 1, r = 1, k = 0.0001: ecomb . . 1427.100h-convergence for σD = 1, γ3, Q = 5, d = 1, r = 1, k = 0.0001: e0, e1 . 1427.101h-convergence for σD = 1, γ3, Q = 5, d = 1, r = 1, k = 0.25: ecomb . . . 1427.102h-convergence for σD = 1, γ3, Q = 5, d = 1, r = 1, k = 0.25: e0, e1 . . . 1437.103k-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1, γ̂ = 10: ecomb . . . . 1437.104k-convergence for σD = 1, γ3, Q = 1, d = 2, r = 1, γ̂ = 10: e0, e1 . . . . 1437.105Infinite domain in 1D: Summary of types . . . . . . . . . . . . . . . . . 148

List of Figures

2.1 Function vǫ and its interpolant Ihvǫ from Example 2.30. . . . . . . . . 29

3.1 Graphs of γ = γ1 and its derivatives. This is the tanh-version of γ. . . . 373.2 Graphs of γ = γ2 and its derivatives. This is the polynomial version of γ. 393.3 Graphs of γ = γ3 and its derivatives. This is the sin(x) + x-version of γ. 40

5.1 A non-unique discrete solution for Q = 4 . . . . . . . . . . . . . . . . . . 705.2 Discrete solution for Q = 4 now unique . . . . . . . . . . . . . . . . . . . 705.3 A non-unique discrete solution for Q = 5 . . . . . . . . . . . . . . . . . . 705.4 Discrete solution for Q = 5 now unique . . . . . . . . . . . . . . . . . . . 70

7.1 Numerical solutions uh(t = 1) for Q = 4 and h = 1/16 . . . . . . . . . . 1077.2 Numerical solution on Ω = (0, 1) . . . . . . . . . . . . . . . . . . . . . . 1457.3 Typical shape of functions G and D for Type I. . . . . . . . . . . . . . 1477.4 Typical shape of functions G and D for Type II. . . . . . . . . . . . . . 1487.5 Typical shape of functions G and D for Type III. . . . . . . . . . . . . 1497.6 Type I: γ(u), γ′(u), γ′′(u), G(u) and D(u) . . . . . . . . . . . . . . . . . 1517.7 Type I: u(x), σ(x), (u+ σ)(x) . . . . . . . . . . . . . . . . . . . . . . . . 1527.8 Type II: γ(u), γ′(u), γ′′(u), G(u) and D(u) . . . . . . . . . . . . . . . . 1537.9 Type II: Zoom to G(u) and D(u) . . . . . . . . . . . . . . . . . . . . . . 1537.10 Type IIa: u(x), σ(x), (u+ σ)(x) . . . . . . . . . . . . . . . . . . . . . . 1547.11 Type IId: u(x), σ(x), (u+ σ)(x), xm(t) . . . . . . . . . . . . . . . . . . 1547.12 Type III: γ(u), γ′(u), γ′′(u), G(u) and D(u) . . . . . . . . . . . . . . . . 1557.13 Type III: Zoom to G(u) and D(u) . . . . . . . . . . . . . . . . . . . . . 1557.14 Type IIIa: u(x), σ(x), (u+ σ)(x) . . . . . . . . . . . . . . . . . . . . . . 1567.15 Type IIIb: u(x), σ(x), (u+ σ)(x), xm(t) . . . . . . . . . . . . . . . . . . 1567.16 Type IIIc: u(x), σ(x), (u+ σ)(x), xm(t) . . . . . . . . . . . . . . . . . . 1577.17 Type IIId: u(x), σ(x), (u+ σ)(x), xm(t) . . . . . . . . . . . . . . . . . . 1577.18 Wave-like numerical solution . . . . . . . . . . . . . . . . . . . . . . . . 1597.19 Colour map for the following plots . . . . . . . . . . . . . . . . . . . . . 1607.20 Domain for 2D example . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.21 Sharp front in 2D: Numerical solution u at t = 1.8 and t = 3.6 . . . . . 1617.22 Sharp front in 2D: Numerical solution u at t = 15 and t = 30 . . . . . . 1627.23 Sharp front in 2D: Numerical solution σ at t = 1.8 and t = 3.6 . . . . . 1637.24 Sharp front in 2D: Numerical solution σ at t = 15 and t = 30 . . . . . . 1647.25 Sharp front in 2D: Numerical solution u+ σ at t = 1.8 and t = 3.6 . . . 1657.26 Sharp front in 2D: Numerical solution u+ σ at t = 15 and t = 30 . . . . 1667.27 Ring-shaped domain in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 1677.28 Ring-shaped domain in 2D: Numerical solution u and σ at t = 20 . . . . 1687.29 Ring-shaped domain in 2D: Numerical solution u+ σ at t = 20. . . . . . 1697.30 Domain for 3D example . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9

LIST OF FIGURES 10

7.31 Sharp front in 3D: Numerical solution u and σ at t = 10 . . . . . . . . . 1717.32 Sharp front in 3D: Numerical solution u+ σ at t = 10. . . . . . . . . . . 172

List of Theorems

Theorem 2.4 (Sobolev embedding) . . . . . . . . . . . . . . . . . . . . . . . . 19Lemma 2.6 (Hilbert space V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Lemma 2.7 (Gelfand triple) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Lemma 2.8 (Green’s formula) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Theorem 2.10 (Unique solution of Laplace equation) . . . . . . . . . . . . . . 22Lemma 2.11 (Inverse Laplacian) . . . . . . . . . . . . . . . . . . . . . . . . . 22Lemma 2.12 (The space Cm([a, b];X)) . . . . . . . . . . . . . . . . . . . . . . 23Lemma 2.13 (An equality for (v̇, v)0) . . . . . . . . . . . . . . . . . . . . . . . 23Lemma 2.14 (The space Wmp (a, b;X)) . . . . . . . . . . . . . . . . . . . . . . 23Lemma 2.15 (Embeddings of time-dependent function spaces) . . . . . . . . . 24Lemma 2.17 (Properties of W(0, T ;V, V ′)) . . . . . . . . . . . . . . . . . . . . 24Proposition 2.18 (Unique solution for ODEs) . . . . . . . . . . . . . . . . . . 25Theorem 2.19 (Existence and uniqueness for linear parabolic problem) . . . . 25Lemma 2.24 (Lagrange basis functions) . . . . . . . . . . . . . . . . . . . . . 27Theorem 2.27 (Inverse Estimates) . . . . . . . . . . . . . . . . . . . . . . . . . 28Lemma 2.29 (Approximation error of interpolation) . . . . . . . . . . . . . . . 28Lemma 2.31 (Scott-Zhang interpolation) . . . . . . . . . . . . . . . . . . . . . 30Lemma 2.33 (Stability of L2- and elliptic projections) . . . . . . . . . . . . . 30Lemma 2.34 (Approximation error for L2- and elliptic projections) . . . . . . 31

Lemma 3.5 (Rescaling the Cohen model) . . . . . . . . . . . . . . . . . . . . . 36Lemma 3.6 (Usual choice for γ) . . . . . . . . . . . . . . . . . . . . . . . . . . 37Lemma 3.7 (Another choice for γ) . . . . . . . . . . . . . . . . . . . . . . . . 38Lemma 3.8 (A third choice for γ) . . . . . . . . . . . . . . . . . . . . . . . . . 38Lemma 3.9 (Difference of γ-values in Lp-norm) . . . . . . . . . . . . . . . . . 40Lemma 3.10 (A bound for C2-functions) . . . . . . . . . . . . . . . . . . . . . 40Lemma 3.11 (Another bound for γ-values) . . . . . . . . . . . . . . . . . . . . 41

Lemma 4.4 (Consequences of triangle inequality) . . . . . . . . . . . . . . . . 44Proposition 4.5 (Basic stability in weak norms) . . . . . . . . . . . . . . . . . 44Proposition 4.6 (Basic stability in stronger norms) . . . . . . . . . . . . . . . 46Lemma 4.8 (Equivalence of weighted max-norms) . . . . . . . . . . . . . . . . 48Lemma 4.9 (Trick with weighted max-norm) . . . . . . . . . . . . . . . . . . . 49Proposition 4.10 (Unique solution of linearised Cohen model) . . . . . . . . . 50Lemma 4.12 (An iteration for the nonlinear problem) . . . . . . . . . . . . . . 52Lemma 4.14 (Bound for difference of iterates) . . . . . . . . . . . . . . . . . . 53Lemma 4.16 (‖σ‖L∞(Ω)-bound of iterate in 1D) . . . . . . . . . . . . . . . . . 55Lemma 4.17 (Banach fixed point theorem for subsets) . . . . . . . . . . . . . 56Proposition 4.18 (Unique solution in 1D) . . . . . . . . . . . . . . . . . . . . . 57

Lemma 5.1 (Taylor estimates) . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11

LIST OF THEOREMS 12

Lemma 5.3 (Properties of discrete inverse Laplacian) . . . . . . . . . . . . . . 62Lemma 5.7 (Matrix form of discretised problem) . . . . . . . . . . . . . . . . 64Lemma 5.8 (Alternative matrix formulation for Q = 3) . . . . . . . . . . . . . 66Lemma 5.9 (Unique discrete solution for Q = 1, 2) . . . . . . . . . . . . . . . 67Lemma 5.12 (Upper bound for BQn ) . . . . . . . . . . . . . . . . . . . . . . . 71Lemma 5.13 (Lower bounds for BQn ) . . . . . . . . . . . . . . . . . . . . . . . 71Proposition 5.14 (Boundedness of discrete solutions using inverse Laplacian) . 72Proposition 5.16 (Boundedness of discrete solutions using u+ σ-terms) . . . . 74Corollary 5.17 (Boundedness of discrete solutions) . . . . . . . . . . . . . . . 76Proposition 5.18 (Bound on differences) . . . . . . . . . . . . . . . . . . . . . 77

Lemma 6.1 (Approximation error for elliptic projection) . . . . . . . . . . . . 79Lemma 6.2 (Bounds involving ξn and θn) . . . . . . . . . . . . . . . . . . . . 80Lemma 6.3 (Error inequalities) . . . . . . . . . . . . . . . . . . . . . . . . . . 81Lemma 6.4 (Intermediate nonlinearity error for methods Q ∈ {2, 5}) . . . . . 85Lemma 6.5 (Intermediate nonlinearity error for methods Q ∈ {1, 4}) . . . . . 87Lemma 6.6 (Extrapolation errors) . . . . . . . . . . . . . . . . . . . . . . . . 88Lemma 6.7 (Nonlinearity errors for methods Q ∈ {1, 2, 4, 5}) . . . . . . . . . . 92Theorem 6.8 (An a priori error estimate) . . . . . . . . . . . . . . . . . . . . . 94Lemma 6.9 (Tentative nonlinearity error for Q = 3) . . . . . . . . . . . . . . . 99

Lemma A.1 (Jensen’s inequality) . . . . . . . . . . . . . . . . . . . . . . . . . 175Lemma A.2 (Continuous Grönwall lemma) . . . . . . . . . . . . . . . . . . . . 175Lemma A.3 (Continuous Grönwall lemma, second form) . . . . . . . . . . . . 176Lemma A.4 (Discrete Grönwall lemma) . . . . . . . . . . . . . . . . . . . . . 176Theorem A.5 (Banach’s fixed point theorem) . . . . . . . . . . . . . . . . . . 177Proposition A.8 (Criterion for convexity/concavity) . . . . . . . . . . . . . . . 177Corollary A.9 (xp is convex if p ≥ 1) . . . . . . . . . . . . . . . . . . . . . . . 177Proposition A.10 (Sub- and superadditivity) . . . . . . . . . . . . . . . . . . . 177Corollary A.12 (xp is superadditive) . . . . . . . . . . . . . . . . . . . . . . . 178Lemma A.13 (Generalising the property of convex functions) . . . . . . . . . 179Proposition A.14 (Equivalence of p-norms in Rn) . . . . . . . . . . . . . . . . 179Theorem A.16 (Hahn-Banach) . . . . . . . . . . . . . . . . . . . . . . . . . . 181Lemma A.17 (An equality with the norm of the dual space) . . . . . . . . . . 181Lemma A.21 (Properties of dual mapping) . . . . . . . . . . . . . . . . . . . . 182Lemma A.22 (Injectivity of dual mapping) . . . . . . . . . . . . . . . . . . . . 182Lemma A.23 (Embedding of dual spaces) . . . . . . . . . . . . . . . . . . . . 182Theorem A.24 (Riesz’ representation theorem) . . . . . . . . . . . . . . . . . 182Proposition A.26 (Gelfand triple) . . . . . . . . . . . . . . . . . . . . . . . . . 183Lemma A.30 (A counter-example) . . . . . . . . . . . . . . . . . . . . . . . . 184Lemma A.31 (Product of symmetric positive definite matrices) . . . . . . . . 184

Corollary B.1 (Amann’s result specialised for the Cohen model) . . . . . . . . 185Theorem B.2 (Amann) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Acknowledgements

I would like to take this opportunity to thank the people who have helped me in manydifferent ways to finish this thesis.

First of all, I thank my supervisor Simon Shaw, without whom this thesis wouldnever have been possible. He always had useful advice and the work with him has beenfun.

I am very grateful to Manu for her invaluable support, especially during the finalmonths of writing. The help from my family also encouraged me a lot.

I would like to express my warmest thanks to Dave Needham, John Leach andJohn Merkin, for valuable discussions and a better understanding of the equations, andAndrew Wathen for pointing out a connection to product approximations.

This work relies heavily on open source software and I am indebted to the developersof the many free software packages like GCC, Gnuplot, LATEX, Python, and many more.Special thanks goes to the authors of ALBERTA finite element toolbox.

Last but not least, I would also like to thank Brunel University and EPSRC fortheir financial support.

13

Chapter 1

Introduction

The aim of this thesis is to give a standard finite element scheme for approximatelysolving a nonlinear PDE system, which was proposed in [CW91] and [CWW95] andwhich we will call the Cohen model. This model was formulated to explain experimentalresults for diffusion in viscoelastic polymers. It combines two departures from theframework of standard Fickian diffusion: firstly, a stress variable is added to give aviscoelastic memory effect and secondly, the evolution of this stress variable contains anonlinearity modelling a phase change of the polymer. For a more detailed description,we refer the reader forward to Sections 3.2 and 3.3.

As a foundation for the results of this thesis, first a variational formulation of themodel is developed and some partial results towards establishing the well-posedness ofthis formulation are given.

Then the finite element scheme is presented, giving a choice of five methods fordiscretising the nonlinearity of the system.

An a priori estimate is proved for four of the methods showing that the discretesolution will converge to the exact solution, provided the exact solution is sufficientlysmooth. Numerical experiments to support the predicted convergence rates are given.This a priori estimate relies on a combination of two approaches to combine the evolu-tion equations of the model. These approaches are introduced in Subsection 4.1.1. Oneof them uses the inverse Laplacian and the other works with the sum of concentrationand stress. An explanation of why only the combination of both approaches seems towork to prove the estimate is given in Subsection 6.4.1.

Finally, the finite element scheme is used to demonstrate some of the behaviour ofthe Cohen model: The concentration can develop sharp fronts and wave-like behaviour.

1.1 Overview of the chapters

Chapter 2 presents some of the mathematical background from functional analysis andfinite element theory which will be needed in the later chapters.

Chapter 3 states the problem that we will examine in this thesis, the Cohen modelfor non-Fickian diffusion. It also provides some background on the physical interpreta-tion and on the history of this model.

Chapter 4 gives a variational formulation of the problem. This formulation is thebasis for the finite element discretisation and error analysis. Here, we will look atquestions of existence and uniqueness of the exact solution.

In Chapter 5 we present the discrete formulation of the problem and prove boundsfor the discrete solution.

14

CHAPTER 1. INTRODUCTION 15

Chapter 6 is the central part of the thesis that contains the main result: the errorestimate that shows the convergence of the discrete scheme.

Chapter 7 gives some numerical results: Tables of convergence confirm the conver-gence rates predicted by the error estimate. Plots of the numerical solution show howthe solution behaves in some special cases.

Chapter 8 is the Conclusion, containing a review of the results of this thesis andsuggestions for further work.

There are also two appendices: Appendix A contains more on the mathematicalbackground and Appendix B gives an interpretation of a result by Amann on existenceof solutions for the Cohen model.

To help the reader find particular items in this thesis, a List of Tables, a List ofFigures and a List of Theorems is included at the beginning, after the Table of Contents.Also, the mathematical symbols that appear more than once can be looked up in theList of Symbols on page 191. Finally, there is an Index, starting on page 196.

Chapter 2

Mathematical Background

This chapter is intended as an introduction to the mathematical framework for thelater chapters. The necessary background from functional analysis and finite elementtheory is presented, firstly to introduce the notation used in this thesis and secondlyto state the results from standard theory that the reader can refer back to later.

Section 2.1 gives functional analytic tools that are used to deal with elliptic problemslike the Laplace equation.

In Section 2.2 further results are given to extend the framework to time-dependentproblems. An abstract existence and uniqueness result for parabolic problems is stated.

Section 2.3 presents a standard finite element discretisation of the Laplace equation.The same discretisation will be applied later to the main problem of this thesis.

Notation

The notation that is used in this thesis is fairly standard. Most symbols and othernotation are defined where they are first needed. For a complete list of notation, werefer the reader to the List of Symbols on page 191 at the end of the thesis. Except forvery basic mathematical notation, every item in that list has a reference to the page orequation in the main text where it is defined.

It is assumed that the reader is familiar with certain basic topics from functionalanalysis and linear algebra, like Banach spaces, Hilbert spaces, etc.

We use C to denote a generic constant, that may have different values in differentplaces. When we want to define constants using explicit expressions we denote themby ci for some i = 1, 2, . . . . The value of these constants remains fixed until the end ofthat chapter.

A shorthand notation used frequently in abstract algebra is: If f : X → Y is amapping from set X into set Y and Z ⊆ X a subset, then we set

f(Z) := {f(z) : z ∈ Z}.

Another convention for this thesis is the use of the multiplication symbol ‘ · ’: It is usedfor normal multiplication and not for the Euclidean inner product. In the few cases,where we need the latter, it is denoted by xT y for x,y ∈ Rn.

Finally, for the notation of integrals, we frequently omit the measure terms such asdΩ, dx from the integrals. For example, we write

∫

Ω f instead of∫

Ω fdΩ.

16

CHAPTER 2. MATHEMATICAL BACKGROUND 17

2.1 Elliptic problems

2.1.1 The domain Ω

This subsection is concerned with the domain for the differential equations. It containsnot only definitions but also assumptions that are made on the domain. Throughoutthis thesis, we only consider domains that satisfy the assumptions made below.

Certain results in this thesis require an even more regular domain. Since theserestrictions will be quite strong, we will introduce a ‘switch’ in Definition 2.3. Usingthat definition, the regularity assumptions can be stated in a concise way. But first westart with the basic assumptions on the domain.

Definition 2.1 (Assumption on domain Ω).Ω is a connected, open, bounded subset of Rd, with dimension d ∈ {1, 2, 3}. MoreoverΩ is assumed to have a regular boundary ∂Ω in the following sense: Either

• ∂Ω is C∞-smooth, or

• Ω is a polyhedral domain, i.e. ∂Ω is a polyhedron, if d = 3, a polygon, if d = 2,or consists of two points, if d = 1.

In both cases, the unit outward normal vector n is defined almost everywhere on theboundary.

To prescribe different types of boundary condition when formulating the PDE sys-tem, we split the boundary of Ω into two parts:

Definition 2.2 (Dirichlet and Neumann boundaries).The boundary of Ω, ∂Ω, consists of two parts, ΓD and ΓN , such that

ΓD ∪ ΓN = ∂Ω, ΓD ∩ ΓN = ∅, (2.1)

where ΓD is closed and has strictly positive (d− 1)-dimensional measure. ΓD is calledthe Dirichlet boundary and ΓN is called the Neumann boundary.

Now we come to the ‘switch’ that was mentioned at the beginning of this subsection.

Definition 2.3 (‘ERq holds’).With the assumptions from Definitions 2.1 and 2.2, we say that ‘ER0 holds’. Further-more, we say that ‘ER1 holds’ if two additional regularity assumptions are satisfied:

• If Ω is a polyhedral domain, then it is convex.

• The whole boundary is a Dirichlet boundary: ∂Ω = ΓD.Note that ER0 is always assumed to hold without stating it explicitly. We will

need the definition of ERq later: If ER1 holds, the solution of the Laplace equationhas a higher regularity (see Theorem 2.10) which is sometimes called elliptic regularity.Also, as a consequence, the L2-error of the elliptic projection has a better bound, seeLemma 2.34. These better bounds will be used for the error estimate in Chapter 6 andwe obtain a higher order of convergence for the discrete schemes.

2.1.2 Sobolev spaces and their embeddings

In this subsection several spaces of functions

v : Ω → R

are defined. However, the Lebesgue and Sobolev spaces defined below are somewhatspecial. Strictly speaking their elements are not functions anymore, but equivalence


classes of functions. Two functions are identified when they differ only on a set ofmeasure 0. For simplicity these equivalence classes are also called ‘functions’. If v is afunction from one of these spaces, and x ∈ Ω is a point, then the expression v(x) mightnot have a well-defined value. On the other hand, the Sobolev embedding theoremwhich is stated later in this subsection, asserts that for certain Sobolev spaces eachfunction has a continuous function in its equivalence class. In these cases the pointwisefunction values v(x) exist.

Defining the Sobolev spaces also involves derivatives of functions. In general thesefunctions might not have classical derivatives, thus so-called distributional derivativeshave to be used, see for example [DL88]. Since the calculation with the distributionalderivatives follows the same rules as for the classical derivatives and since both deriva-tives coincide whenever the classical derivative exists, we do not distinguish betweenthem.

In this subsection we state, along with the definitions, several continuous embed-dings of the function spaces. For a definition of continuous embeddings, we refer thereader to the appendix, Definition A.18.

Space of Continuous Functions

First of all define C0(Ω̄) as the space of continuous functions Ω̄ → R. This space is aBanach space, when equipped with the norm

‖v‖C0(Ω̄) := maxx∈Ω̄

|v(x)|, (2.2)

for any v ∈ C0(Ω̄), see for example [BG87a, 2.3A]. Note that the maximum in (2.2) isattained, by the Weierstrass theorem, since we assume Ω to be bounded.

Lebesgue spaces

For any 1 ≤ p ≤ ∞, the norm of the Lebesgue spaces is defined by

‖v‖Lp(Ω) :=(∫

Ω|v|p dΩ

)1/p

for 1 ≤ p


Sobolev Hilbert spaces

Set, for m ∈ N,Hm(Ω) := Wm2 (Ω),

and define the inner product

(v,w)Hm(Ω) :=∑

|α|≤m

∫

Ω∂αv · ∂αw dΩ,

then Hm(Ω) together with (·, ·)Hm(Ω) is a Hilbert space. Note that H0(Ω) = L2(Ω).As for any Hilbert space, the inner product satisfies the Cauchy-Schwarz inequality:

(v,w)Hm(Ω) ≤ ‖v‖Hm(Ω) · ‖w‖Hm(Ω) for v,w ∈ Hm(Ω). (2.5)

To have a shorter notation for the L2-norm and inner product, we set

(·, ·)0 := (·, ·)L2(Ω),‖·‖0 := ‖·‖L2(Ω).

(2.6)

This notation is used for example in [Bra97].

The Product space L2(Ω)2

We will sometimes use the space L2(Ω)2 = L2(Ω) × L2(Ω) with norm

‖(v,w)‖L2(Ω)2 :=(

‖v‖2L2(Ω) + ‖w‖2L2(Ω)

)1/2, (2.7)

for any v,w ∈ L2(Ω). As L2(Ω) is a Hilbert space it follows that L2(Ω)2 is also aHilbert space, see also [BG87b, Example 1.15].

Sobolev embeddings

As was mentioned before, Sobolev spaces consist of equivalence classes of functions.However, the elements of certain Sobolev spaces can be identified with continuousfunctions and the Sobolev embedding theorem gives conditions under which this is thecase. We cite a version of the Sobolev embedding theorem from [BS94]:

Theorem 2.4 (Sobolev embedding).Let k be a positive integer, m an integer with m < k and let p be a real number in therange 1 ≤ p d/p when p > 1,

where d is the dimension of Ω. Then there is a constant C such that for all u ∈W kp (Ω)

‖u‖W m∞

(Ω) ≤ C‖u‖W kp (Ω).

Moreover, there is an m times continuously differentiable function in the L∞(Ω) equiv-alence class of u.

Proof. This is [BS94, Theorem (1.4.6) and Corollary (1.4.7)].


2.1.3 Laplace equation

In this subsection the space V is defined which will be used for the variational for-mulation of the Cohen model. The idea of the variational formulation is introducedfor the Laplace equation, as this is the building block for the time-dependent problem.The theory can be found for example in [OD96, Section 6.6]. For a more rigorousformulation using trace theorems, see [DL88, Section VII.2.2.1].

The Laplace equation, as a PDE on a bounded domain is:

Problem 2.5 (Laplace equation as PDE).Given f ∈ L2(Ω) and g ∈ L2(ΓN ), find u ∈ H2(Ω) such that

−∆u = f in Ω, (2.8a)u = 0 on ΓD, (2.8b)

(∇u)T n = g on ΓN . (2.8c)In order to give a variational formulation for this problem, first the space V is

defined and some of its properties are stated: Note that a rigorous definition of thespace V in (2.9) and also of the Neumann boundary condition (2.8c) needs a tracetheorem, see for example [DL88, IV.4.3].

Lemma 2.6 (Hilbert space V ).Define the space V by

V := {v ∈ H1(Ω) : v = 0 on ΓD}, (2.9)and set for any v,w ∈ V :

(v,w)V := (∇v,∇w),‖v‖V :=

√

(v, v)V .(2.10)

Then (V, (·, ·)V ) is a Hilbert space. Moreover, on V the V -norm is equivalent to theH1-norm: there is a constant CV > 0 such that

C−1V ‖v‖H1(Ω) ≤ ‖v‖V ≤ ‖v‖H1(Ω) ∀ v ∈ V. (2.11)Proof. We just give an outline of the proof: By [DL88, Section VII.2.2.1], the spaceV is a closed (w.r.t ‖·‖H1(Ω)) subspace of H1(Ω). Hence V together with ‖·‖H1(Ω) isa Banach space. Furthermore, it is easily seen that (·, ·)V is a symmetric, positive-semidefinite bilinear form and hence that ‖·‖V is a seminorm. (Actually ‖·‖V is the socalled H1-seminorm on the space H1(Ω).) Thus if (2.11) can be shown, then we wouldfirstly get that ‖·‖V is a norm and that (·, ·)V is positive definite and thus an innerproduct on V . The completeness of V with respect to ‖·‖V would also follow from thecompleteness of V with respect to ‖·‖H1(Ω) proving that V is a Hilbert space. So theonly thing left to prove is (2.11).

Now, the right half of the equivalence (2.11) is an immediate consequence of thedefinition of the H1-norm, but the left half is the difficult part: This is Poincaré’sinequality, which has an easy proof if ∂Ω = ΓD, see [DL88, IV.7.1, Proposition 1].For the general case where ΓD 6= ∂Ω, the proof makes use of the fact that H1(Ω) iscompactly embedded in L2(Ω), see [DL88, IV.7.2, Proposition 2 and Remark 4].

If ER1 holds, and hence the Dirichlet boundary consists of the whole boundary, thenthe space V is also denoted as H10 (Ω), see for example [Eva98].

We now come to the connection between the two spaces V and L2(Ω), they formpart of a so-called Gelfand triple. For more background on this subject, including thedefinitions of the dual space, the duality pairing and Gelfand triples, see Section A.4in the Appendix.


Lemma 2.7 (Gelfand triple).V is continuously and densely embedded in L2(Ω), i.e.

‖v‖0 ≤ CV ‖v‖V ∀v ∈ V, (2.12)

where CV > 0 is the constant from (2.11). Furthermore, V,L2(Ω), V′ form a Gelfand

tripleV →֒ L2(Ω) →֒ V ′.

Let 〈·, ·〉 : V ′ × V → R denote the duality pairing of V with V ′. Then

〈w, v〉 = (w, v)0 ∀w ∈ L2(Ω), v ∈ V, (2.13)

and‖v‖V ′ ≤ CV ‖v‖0 ∀v ∈ L2(Ω). (2.14)

Proof. Again we give just an outline of the proof: By the definition of H1(Ω) we havethat H1(Ω) →֒ L2(Ω) and by (2.11) we get V →֒ H1(Ω). The superposition of these twocontinuous embeddings gives the continuous embedding V →֒ L2(Ω). By the definitionof the H1-norm we have ‖v‖0 ≤ ‖v‖H1(Ω) for v ∈ V , and together with (2.11) thisimplies (2.12).

The embedding of V in L2(Ω) is also dense, since the embedding of H10 (Ω) in L2(Ω)

is. Thus Proposition A.26 can be used to obtain the Gelfand triple and (2.13).It remains to confirm the constant in (2.14): Let v ∈ L2(Ω). Then

‖v‖V ′ = sup06=w∈V

| 〈v,w〉 |‖w‖V

= sup06=w∈V

| (v,w)0 |‖w‖V

≤ sup06=w∈V

‖v‖0‖w‖0‖w‖V

≤ sup06=w∈V

‖v‖0 CV ‖w‖V‖w‖V

= CV ‖v‖0.

Now, having defined the space V we can reformulate Problem 2.5 in a variationalform. Green’s formula is the tool that is used for this.

Lemma 2.8 (Green’s formula).For sufficiently regular functions u, v, we have

∫

Ω(−∆u)v =

∫

Ω(∇u)T∇v −

∫

∂Ω(∇u)T nv.

Supposing that the solution u to Problem 2.5 and an arbitrary function v ∈ V aresufficiently smooth, we can multiply (2.8a) by v and integrate over the domain, thenapply Green’s formula, and finally use the Neumann boundary condition (2.8c) to arriveat (2.15) given below. This new formulation makes sense not only for these sufficientlysmooth functions but even for functions which are less regular than in Problem 2.5.This leads to the following variational formulation:

Problem 2.9 (Variational form of the Laplace equation).Given f ∈ L2(Ω) and g ∈ L2(ΓN ), find u ∈ V such that

(u, v)V = 〈L, v〉 ∀v ∈ V, (2.15)

where L ∈ V ′ is defined as

〈L, v〉 := (f, v)0 + (g, v)ΓN for v ∈ V.


The standard result of existence and uniqueness for Problem 2.9 is:

Theorem 2.10 (Unique solution of Laplace equation).If L ∈ V ′ then (2.15) has a unique solution u ∈ V and

‖u‖V = ‖L‖V ′ .Furthermore, if ER1 holds and if L ∈ L2(Ω) then u ∈ H2(Ω) and

‖u‖H2(Ω) ≤ C‖L‖0.Proof. The first part follows immediately from the Riesz representation theorem A.24.For the second part see [EG04, Theorem 3.12] for polygonal domains and see [RR04,Theorem 9.53] for C∞-domains.

For getting the higher regularity u ∈ H2(Ω) in Theorem 2.10, it is necessary tomake assumptions similar to ER1:

• If in two dimensions the domain is polygonal but not convex, then there is aso called re-entrant corner, i.e. a corner with interior angle > π. In that casethe solution u has a singularity at this corner and u /∈ H2(Ω), see [EG04, Re-mark 3.13(ii)].

• If the type of boundary condition changes on connected parts of the boundary,then the solution might also not be in H2(Ω) anymore, for an example, see [EG04,Remark 3.11(ii)].

2.1.4 Inverse Laplacian

Another concept sometimes used for systems of partial differential equations is theinverse Laplacian, see for example [BGN03].

For v ∈ V ′ the inverse Laplacian Gv is defined as the unique solution to the varia-tional problem

(Gv,w)V = 〈v,w〉 ∀w ∈ V. (2.16)Note that G is linear and has the following properties:Lemma 2.11 (Inverse Laplacian).G : V ′ → V is an isometric isomorphism, in particular:

‖Gv‖V = ‖v‖V ′ ∀v ∈ V ′. (2.17)Furthermore, if ER1 holds then the inverse Laplacian has a strong stability property:There is C > 0 such that

‖Gv‖H2(Ω) ≤ C‖v‖0 ∀ v ∈ L2(Ω). (2.18)Proof. Use the Riesz representation theorem A.24 and Theorem 2.10.

2.2 Time dependent problems

The model problem that is examined in this thesis is time-dependent, hence in thissection some standard spaces and tools from functional analysis to deal with timedependent problems are stated.

The time t will always be taken from a bounded interval,

t ∈ [0, T ],for some time end point T < ∞. The time derivatives are denoted using dots. Wewrite v̇, v̈,

...v for the first, second and third derivative of v with respect to time t.


2.2.1 Spaces for time dependent problems

This subsection gives an introduction to the function spaces that are needed to deal withtime-dependent problems. Almost always, the time derivatives that appear in physicalequations have a different form than the space derivatives. Therefore, the parametertime cannot be treated as just an extra space dimension. The function spaces usuallyneed different regularity with respect to space than with respect to time.

Let J ⊆ R be a time interval. We want to consider functions that depend on bothtime and space, i.e. functions v : J × Ω → R. In order to define function spaces forthese functions, we reuse the definitions of function spaces from Section 2.1.2. Let Xbe a Banach space of space-dependent functions Ω → R, for example one of those ofSection 2.1.2. We identify the functions v : J → X with the functions v : J × Ω → Rby writing

v(t)(x) = v(t,x).

Let a < b. We define the function spaces either on the closed time interval t ∈ [a, b] oron the open counterpart t ∈ (a, b):Lemma 2.12 (The space Cm([a, b];X)).Let (X, ‖·‖X ) be a Banach space, m ∈ N and [a, b] a bounded interval. Then Cm([a, b];X)is defined as the space of m times continuously differentiable functions from [a, b] intoX. This space together with the norm

‖v‖Cm([a,b];X) :=m∑

i=0

supt∈[a,b]

‖ ∂i∂tiv(t)‖X (2.19)

is a Banach space.

Proof. For the definition and statement of the properties of Cm([a, b];X), see [BG87a,2.10B].

For C0([a, b];X) we also write C([a, b];X) and in this case the definition of the normsimplifies to:

‖v‖C([a,b];X) := maxt∈[a,b]

‖v(t)‖X . (2.20)

Later we will frequently need the next equality for functions in C1([0, T ];L2(Ω)):

Lemma 2.13 (An equality for (v̇, v)0).For any v ∈ C1([0, T ];L2(Ω)):

(v̇, v)0 =1

2

d

dt‖v‖20. (2.21)

Proof. This follows from the definition of the L2 inner product, since

d

dt‖v‖20 =

d

dt

∫

Ωv2 =

∫

Ω2v̇v = 2(v̇, v)0.

The spaces Wmp (a, b;X)

The time dependent version of the Sobolev spaces are:

Lemma 2.14 (The space Wmp (a, b;X)).Let m ∈ N and 1 ≤ p ≤ ∞. Then Wmp (a, b;X) is defined as the space of functions(a, b) → X such that their X-norm lies in Wmp ((a, b)). For v ∈Wmp (a, b;X) set

w :

{

(a, b) → Rt 7→ ‖v(t)‖X ,


and use this to define the norm

‖v‖W mp (a,b;X) := ‖w‖W mp ((a,b)).

When equipped with this norm, Wmp (a, b;X) is a Banach space.

Proof. See [BG87a, 2.10D].

As before for the space dependent functions, the special cases m = 0 and p = 2 arealso denoted differently:

Lp(a, b;X) := W0p (a, b;X), H

m(a, b;X) := Wm2 (a, b;X).

The norms for these spaces are also defined in this way.

Lemma 2.15 (Embeddings of time-dependent function spaces).i) Let X, Y be Banach spaces such that X →֒ Y . Then for any m ∈ N and 1 ≤ p ≤ ∞,

Wmp (a, b;X) →֒Wmp (a, b;Y ),Cm([a, b];X) →֒ Cm([a, b];Y ).

ii) Let X be a Banach space. If l,m ∈ N and 1 ≤ p, q ≤ ∞ are such that W lp((a, b)) →֒Wmq ((a, b)) then

W lp(a, b;X) →֒Wmq (a, b;X).

Proof. This follows from the definitions. See also [BG87a, 2.10B, 2.10D].

As an example of the use of this lemma, observe that (since the time-interval (0, T )is one-dimensional), the Sobolev embedding Theorem 2.4 implies that H2((a, b)) →֒W 1∞((a, b)), and hence

H2(0, T ;L2(Ω)) →֒ W 1∞(0, T ;L2(Ω)), (2.22)

an embedding that will come to use later in Chapter 6.

The space W(0, T ;V, V ′)There is another important space of time dependent functions, which we will define now.It is the space in which an important class of time-dependent problems has a uniquesolution, see later in Theorem 2.19. The definition is taken from [DL88, XVIII.1.2Definition 4] and [Wlo87, Definition 25.3].

Definition 2.16 (The space W(0, T ;V, V ′)).Let V →֒ L2(Ω) →֒ V ′ be the Gelfand triple from Lemma 2.7. Define

W(0, T ;V, V ′) := {u : u ∈ L2(0, T ;V ), u̇ ∈ L2(0, T ;V ′)}.

Associated with this space is the norm given by

‖u‖W(0,T ;V,V ′) :=(

‖u‖2L2(0,T ;V ) + ‖u̇‖2L2(0,T ;V ′)

)1/2.

Lemma 2.17 (Properties of W(0, T ;V, V ′)).i) W(0, T ;V, V ′) is a Hilbert space.ii) If w ∈ W(0, T ;V, V ′), then w has a function in its equivalence class, which iscontinuous in time and

W(0, T ;V, V ′) →֒ C([0, T ];L2(Ω)). (2.23)


iii) Let w ∈ W(0, T ;V, V ′). Then

〈ẇ, w〉 = 12

d

dt‖w‖20. (2.24)

iv) Let w ∈ W(0, T ;V, V ′). Then

〈ẇ,Gw〉 = 12

d

dt‖w‖2V ′ , (2.25)

where G is the inverse Laplacian, defined in Subsection 2.1.4.

Proof. i) see [DL88, XVIII.1.2 Proposition 6] or [Wlo87, Theorem 25.4].ii) See [Wlo87, Theorem 25.5]. Similar statements can be found in [DL88, XVIII.1.2Theorem 1 and Remark 4] and [Eva98, 5.9 Theorem 3].iii) See [Tem97, Chapter II, Lemma 3.2].iv) First we get from (2.16) and (2.17), for any w ∈ W(0, T ;V, V ′),

(w,Gw)0 = (Gw,Gw)V = ‖Gw‖2V = ‖w‖2V ′ .

To continue, for any w ∈ L2(0, T ;V ) with ẇ ∈ L2(0, T ;V ), we have that

d

dt(w,Gw)0 = (ẇ,Gw)0 + (w,Gẇ)0 = 2(ẇ,Gw)0 = 2〈ẇ,Gw〉,

since (w,Gẇ)0 = (Gw,Gẇ)V = (Gẇ,Gw)V = (ẇ,Gw)0. By combining the aboveequations, we get (2.25), if ẇ ∈ L2(0, T ;V ). The result extends by density to anyw ∈ W(0, T ;V, V ′).

2.2.2 Unique solution of ODEs

In Chapter 4 we will need an an existence and uniqueness result for ordinary differentialequations. As we do not need the full generality of the Picard-Lindelöf theorem, wequote a result for the special case of linear ODEs:

Proposition 2.18 (Unique solution for ODEs).Let Y be a Banach space, A ∈ C([0, T ];L (Y, Y )), b ∈ C([0, T ];Y ) and y0 ∈ Y . Thenthe linear inhomogeneous initial value problem

ẏ = A(t)y + b(t),

y(0) = y0,

has a unique solution y ∈ C1([0, T ];Y ).A proof can be found for example in [Ama90, Theorem (7.9) and Remark (7.10) c)].

2.2.3 Linear parabolic equations

Another result of existence and uniqueness that will be needed in Chapter 4 is thefollowing theorem. For the proof see for example [Wlo87, §26] or [RR04, Theorem 11.3].Theorem 2.19 (Existence and uniqueness for linear parabolic problem).Suppose that L ∈ L2(0, T ;V ′) and ŭ ∈ L2(Ω). Then the problem

〈u̇, v〉 + (u, v)V = 〈L, v〉 ∀v ∈ V ∀t ∈ (0, T ) a.e. (2.26a)u(0) = ŭ (2.26b)

has a unique solution u ∈ W(0, T ;V, V ′).


Remark 2.20.Note that the initial condition (2.26b) has sense for u ∈ W(0, T ;V, V ′), i.e. u(0) iswell-defined, since there is a continuous function in the equivalence class of u:

u ∈ C([0, T ];L2(Ω)),

by (2.23).

2.3 Space discretisation

In this section we present some elements from standard finite element theory and againdefine the notation used in the later chapters.

2.3.1 Mesh and finite element spaces

In Definition 2.1 Ω was required to have either a smooth or a polyhedral boundary.However for the space discretisation used in this thesis the domain is always assumedto be polyhedral. This is done for simplicity.

Assumption 2.21.For any h ∈ (0, ĥ], construct a mesh Eh,which is a finite set of elements that satisfiesthe following conditions:

• Each finite element is a non-empty open set in Rd and E ∈ Eh is a simplex , i.e.an interval if d = 1, a triangle if d = 2 or a tetrahedron if d = 3.

• The elements form a partition of the domain Ω, i.e. for any h⋃

E∈EhĒ = Ω̄,

E1 ∩ E2 = ∅ ∀E1, E2 ∈ Eh with E1 6= E2.

• The meshes are geometrically conformal, there are no “hanging nodes”: If forelements E1, E2 ∈ Eh the set Ē1 ∩ Ē2 has positive (d − 1)-dimensional measure,then it is an edge (d = 2) or a face (d = 3) of both E1 and E2. (Compare [EG04,Definition 1.55, Remark 1.56].)

• The type of boundary condition does not change on an edge or face: If f is anedge (d = 2) or a face (d = 3) of an element, then f ⊆ ∂Ω implies either f ⊆ ΓDor f ⊆ ΓN .

• The element diameters are bounded by h, i.e.

∀h ∀E ∈ Eh hE ≤ h,

where hE is the diameter of element E:

hE := supx,y∈E

|x − y|.

• The mesh family is quasi-uniform, i.e. there is a constant τ > 0 such that

∀h ∀E ∈ Eh ρE ≥ τh,

where ρE is the diameter of the largest ball that can be inscribed in E.


Definition 2.22 (Finite element spaces).Let Eh be a mesh for a certain h > 0. Let also r ≥ 1. The finite element spaces W hand V h are defined by

W h := {v ∈ C0(Ω̄) : ∀E ∈ Eh v|E ∈ Pr(E)},V h := {v ∈W h : v|ΓD = 0},

(2.27)

where Pr(E) is the set of polynomial functions in d variables of total degree less thanor equal to r on E ⊆ Rd.Definition 2.23 (Lagrange nodes).For E ∈ Eh let aE,i, i ∈ {0, . . . , d} denote the corner points of the simplex E. The setof local Lagrange nodes of E is defined as

LN rE :={

d∑

i=0

liraE,i : l0, l1, . . . , ld ∈ {0, . . . , r} ∧

d∑

i=0

li = r

}

,

where each tuple (l0, l1, . . . , ld) with sum equal to r gives rise to one Lagrange node.Note that the tuples ( l0r ,

l1r , . . . ,

ldr ) form the barycentric coordinates of the Lagrange

nodes, for more information, see [EG04, Section 1.2.3]. The set of global Lagrange nodesis then defined by

LN r :=⋃

E∈EhLN rE

Let NW := |LN r| be the total number of Lagrange nodes and NV := |LN r \ ΓD|the number of Lagrange nodes which are not on the Dirichlet boundary. Denote theLagrange nodes by aj and enumerate them in such a way that

LN r = {aj : j = 1, . . . ,NW }, (2.28)and that for j ∈ {1, . . . , NW },

aj ∈ ΓD ⇐⇒ j > NV .Lemma 2.24 (Lagrange basis functions).With the definitions from above:

1. For any j = 1, . . . , NW , the conditions

φj(ai) = δij ∀ i ∈ {1, . . . ,NW } (2.29)uniquely define a function φj ∈W h.

2. The set {φi : i = 1, . . . ,NW } forms a basis of W h and {φi : i = 1, . . . ,NV }forms a basis of V h.

3. W h is H1(Ω)-conformal and V h is V -conformal:

W h ⊆ H1(Ω), V h ⊆ V.

Proof. This is standard finite element theory, see for example [EG04, Proposition 1.34and Section 1.4.5]. For the conformality, see [EG04, Proposition 1.74].

Remark 2.25.Although the finite element space depends not only on h but also on the polynomialdegree r, we do not write it as another index V h,r. It is always assumed that r is fixed.

Remark 2.26 (Why Lagrange basis functions?).The specific choice of Lagrange basis functions here is made purely for practical rea-sons: the ALBERTA library, which was chosen for the numerical experiments, uses theLagrange basis functions up to the polynomial degree of r = 4, see later in Section 7.2.


2.3.2 Inverse estimates

Since in the finite dimensional space V h all norms are equivalent, any norm on V h

can be bounded by any other norm. However the equivalence constants can dependon h. In this subsection so-called inverse estimates are stated, which give the form ofh-dependence in some cases. The bounds are mostly in the opposite direction, whencompared to the Sobolev embedding, Theorem 2.4.

Theorem 2.27 (Inverse Estimates).Suppose that 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and 0 ≤ m ≤ l, such that V h ⊆W lp(Ω) ∩Wmq (Ω).Then there is a constant C > 0, independent of h such that

‖v‖W lp(Ω) ≤ C hm−l+min(0, dp−

dq )‖v‖W mq (Ω),

for all v ∈ V h.

Proof. This is a special case of [Cia02, Theorem 3.2.6] (or [BS94, Theorem (4.5.11)],which has the technical restriction of h < 1).

As the finite element space V h contains functions which are in general not contin-uously differentiable across element boundaries, the restriction V h ⊆ W lp(Ω) ∩Wmq (Ω)is quite severe. Therefore, the inverse estimates are usually stated in a more generalform, where the W lp-norm and W

mq -norm is applied on each element and not on the

whole domain. For the sake of simplicity and because we do not need the general form,the inverse estimates are stated the way they are.

For later, we note some special cases of the inverse estimates:

‖v‖L∞(Ω) ≤ Ch−d/2 ‖v‖0 ∀ v ∈ V h.

‖v‖V ≤ Ch−1 ‖v‖0 ∀ v ∈ V h. (2.30)

2.3.3 Interpolation

Nodal interpolation is used for one of the five discrete schemes that are considered inChapter 5.

Definition 2.28 (Nodal interpolation).The nodal interpolation operator Ih : C0(Ω̄) →W h is defined by

Ihv :=

NW∑

j=1

v(aj)φj , (2.31)

for any v ∈ C0(Ω̄).Note that as a consequence of this definition, nodal interpolation preserves homo-

geneous Dirichlet boundary conditions: Ih(C0(Ω̄) ∩ V ) ⊆ V h.We state a standard result of the approximation error of nodal interpolation, i.e. of

how close the interpolant is to the original function:

Lemma 2.29 (Approximation error of interpolation).Suppose that d2 − 1 < l ≤ r. Then for any v ∈ H l+1(Ω) ∩ V :

‖v − Ihv‖0 ≤ Chl+1‖v‖Hl+1(Ω),‖v − Ihv‖V ≤ Chl‖v‖Hl+1(Ω).


Proof. This is a special case of [EG04, Corollary 1.109, Example 1.106]. Observe that lhas to be chosen large enough so that H l+1(Ω) is continuously embedded in C0(Ω̄).

One problem with the nodal interpolation operator is its lack of stability withrespect to the L2-norm. To demonstrate this, we give an example.

Example 2.30 (About stability of the interpolation operator).We are going to show the following:There is no C > 0, independent of h, such that

‖Ihv‖0 ≤ C‖v‖0 ∀v ∈ C∞[−1, 1].

Note that we choose the space of infinitely differentiable functions here. Since there isno stability in this space, as we will see, we cannot have stability in any bigger, lessregular space containing C∞[−1, 1].

To give the proof, let Ω = (−1, 1) in 1D be discretised with a mesh with equidistantnodes: . . . ,−2h,−h, 0, h, 2h, . . . using piecewise polynomials, r = 1. For each ǫ ∈ (0, h)choose a function vǫ ∈ C∞[−1, 1] such that 0 ≤ vǫ(x) ≤ 1, vǫ(0) = 1 and vǫ(x) = 0 for|x| ≥ ǫ. Such a function exists, a possible choice would be, for example,

vǫ(x) :=

{

exp(1 − 1/(1 − (x/ǫ)2)) if |x| < ǫ,0 otherwise.

See Figure 2.1 for a sketch of vǫ and its interpolant Ihvǫ. Now since ǫ < h we have that

1

0

ǫ h0−ǫ−hx

Figure 2.1: Function vǫ and its interpolant Ihvǫ from Example 2.30.

‖Ihvǫ‖0 =(

2

∫ h

0

(

1 − xh

)2dx

)1/2

=√

2h/3,

but on the other hand ‖vǫ‖0 ≤√

2ǫ. Thus for each C > 0 there is an ǫ > 0 such‖Ihvǫ‖0 > C‖vǫ‖0, which is what we wanted to prove.An equivalent statement to the above is: There is no C > 0 such that

‖v − Ihv‖0 ≤ C‖v‖0 ∀v ∈ C∞[−1, 1].

Scott-Zhang interpolation

Because of the main drawback of the nodal interpolation Ih, i.e. that it can onlybe applied to continuous functions, other types of interpolation operators have beendefined and studied. Two such examples are the Clément operator and Scott-Zhanginterpolation, see [EG04, Section 1.6].


Here, we quote properties of the Scott-Zhang interpolation, without giving a defi-nition of it. This type of interpolation will only be needed in this chapter; in the nextsubsection we will use it to prove some properties of the L2- and elliptic projections.

Lemma 2.31 (Scott-Zhang interpolation).There is a projection operator SZh : H1(Ω) → W h which preserves homogeneous bound-ary conditions such that SZh(V ) ⊆ V h. This interpolation operator is stable in H1(Ω),

‖SZhv‖H1(Ω) ≤ C‖v‖H1(Ω), (2.32)

and its approximation properties are, for l ∈ {0, . . . , r},

‖v − SZhv‖0 ≤ Chl+1 ‖v‖Hl+1(Ω) ∀ v ∈ H l+1(Ω), (2.33)‖v − SZhv‖H1(Ω) ≤ Chl ‖v‖Hl+1(Ω) ∀ v ∈ H l+1(Ω). (2.34)

Proof. This is a specialisation of [EG04, Lemma 1.130].

2.3.4 L2-projection and elliptic projection

This subsection defines two projection operators that map into V h, namely the L2-projection and the elliptic projection. Both are orthogonal projections from a biggerspace onto the subspace V h. Some stability and approximation properties of theseoperators are included.

The L2-projection will be used for calculating the discrete initial solution, see laterin Section 5.2. The elliptic projection will be needed for proving the error estimate inChapter 6.

Definition 2.32 (L2- and elliptic projection).For v ∈ V define the elliptic projection ΠhV v ∈ V h via

(ΠhV v,w)V = (v,w)V ∀w ∈ V h.

Similarly, for v ∈ L2(Ω) define the L2-projection Πh0v ∈ V h via

(Πh0v,w)0 = (v,w)0 ∀w ∈ V h.

Note that the form of the elliptic projection is usually problem dependent and comesfrom the form of the space derivatives of the problem under study. Thus the ellipticprojection is sometimes defined using the full H1 inner product (·, ·)V +(·, ·)0 instead ofonly the inner product (·, ·)V of V , see for example [EG04]. The next lemma is [EG04,Lemma 1.113], modified only to have the V -norm instead of the (equivalent) H1-norm:

Lemma 2.33 (Stability of L2- and elliptic projections).There is C > 0, independent of h, such that

‖Πh0v‖0 ≤ ‖v‖0 ∀v ∈ L2(Ω),‖ΠhV v‖V ≤ ‖v‖V ∀v ∈ V,‖Πh0v‖V ≤ C‖v‖V ∀v ∈ V.

Proof. Using the definition of the L2-projection and the Cauchy-Schwarz inequality, weget

‖Πh0v‖20 = (Πh0v,Πh0v)0 = (v,Πh0v)0 ≤ ‖v‖0 · ‖Πh0v‖0.Dividing by ‖Πh0v‖0 yields the first inequality.

Analogously, for the elliptic projection,

‖ΠhV v‖2V = (ΠhV v,ΠhV v)V = (v,ΠhV v)V ≤ ‖v‖V · ‖ΠhV v‖V ,


which gives the second inequality.The proof of the third inequality follows [EG04, Exercise 1.17], for another proof,

see [Bra97, Corollary 7.8]. Let v ∈ V . For any z ∈ V h we have by the triangle inequality

‖Πh0v‖V ≤ ‖Πh0v − z‖V + ‖z‖V . (2.35)

The choice of z depending on v will be made later. To obtain a bound for the term‖Πh0v − z‖V , we first find a bound for ‖Πh0v − z‖0:

‖Πh0v − z‖20 = (Πh0v − z,Πh0v − z)0 = (Πh0v,Πh0v − z)0 − (z,Πh0v − z)0= (v,Πh0v − z)0 − (z,Πh0v − z)0 = (v − z,Πh0v − z)0≤ ‖v − z‖0 · ‖Πh0v − z‖0,

whence‖Πh0v − z‖0 ≤ ‖v − z‖0.

Since Πh0v − z ∈ V h we can use the inverse estimate (2.30), which gives

‖Πh0v − z‖V ≤ Ch−1‖Πh0v − z‖0.

Combining all these with (2.35) gives

‖Πh0v‖V ≤ Ch−1‖v − z‖0 + ‖z‖V . (2.36)

The remaining problem is to choose a function z ∈ V h for which these terms can bebounded by ‖v‖V . Choosing the nodal interpolant z = Ihv would require that v iscontinuous in order to apply Lemma 2.29 and this is the case only for d = 1. Thereforethis choice for z would not be useful in general. The Scott-Zhang interpolant SZhv,however, has the necessary properties: The stability

‖SZhv‖V ≤ C‖v‖V ,

follows from (2.32) since v,SZhv ∈ V and because of the equivalence of V -norm andH1-norm (2.11). The approximation property is a consequence of (2.33),

‖v − SZhv‖0 ≤ Ch‖v‖V .

Thus, choosing z = SZhv and using stability and approximation in (2.36), we get

‖Πh0v‖V ≤ C‖v‖V ,

which is what we wanted to prove.

Lemma 2.34 (Approximation error for L2- and elliptic projections).Suppose that ERq holds and that 0 ≤ l ≤ r. There is C > 0, independent of h, suchthat

‖v − Πh0v‖0 ≤ Chl+1 ‖v‖Hl+1(Ω) ∀v ∈ H l+1(Ω), (2.37)‖v − ΠhV v‖0 ≤ Chl+q ‖v‖Hl+1(Ω) ∀v ∈ V ∩H l+1(Ω), (2.38)‖v − ΠhV v‖V ≤ Chl ‖v‖Hl+1(Ω) ∀v ∈ V ∩H l+1(Ω). (2.39)

Note that the ‘missing’ approximation property

‖v − Πh0v‖V ≤ Chl ‖v‖Hl+1(Ω) ∀v ∈ V ∩H l+1(Ω),

holds as well, see [EG04, Proposition 1.134], but it is not needed for this thesis.


Proof of Lemma 2.34. The proof is very similar to the one of [EG04, Proposition 1.134],it only needs to be adapted for the different definition of the elliptic projection and theuse of the V -norm instead of the H1-norm. We start by proving (2.39). First observethat the definition of the elliptic projection directly implies this orthogonality equation:

(v − ΠhV v,w)V = 0 ∀w ∈ V h. (2.40)

Using this we get the following chain of inequalities: For any vh ∈ V h,

‖v − ΠhV v‖2V = (v − ΠhV v, v − ΠhV v)V = (v − ΠhV v, v − vh)V≤ ‖v − ΠhV v‖V · ‖v − vh‖V .

Thus‖v − ΠhV v‖V ≤ ‖v − vh‖V ∀ vh ∈ V h.

Choosing vh := SZhv and using (2.34) yields (2.39).The proof of (2.37) is directly analogous, using (·, ·)0 and ‖·‖0 instead of (·, ·)V and

‖·‖V and finally using (2.33).Now to the proof of (2.38): For q = 0, this is a direct consequence of (2.39) and

(2.12). Therefore suppose now that q = 1. We use the technique of the Aubin-NitscheLemma for the proof (see for example [EG04, Section 2.3.4]). Let v ∈ V and sete = v − ΠhV v. Define z ∈ V as the solution to the dual problem

(z,w)V = (e,w)0 ∀w ∈ V. (2.41)

Then by Theorem 2.10 and using the elliptic regularity ER1, we have that z ∈ H2(Ω)and

‖z‖H2(Ω) ≤ C‖e‖0. (2.42)Now

‖e‖20 = (e, e)0(2.41)= (z, e)V

(2.40)= (z − ΠhV z, e)V ≤ ‖z − ΠhV z‖V · ‖e‖V

(2.39)

≤ Ch‖z‖H2(Ω) · hl‖v‖Hl+1(Ω)(2.42)

≤ Chl+1‖e‖0 · ‖v‖Hl+1(Ω),

whence‖e‖0 ≤ Chl+1‖v‖Hl+1(Ω),

which is (2.38).

Chapter 3

The Cohen Model for

Non-Fickian Diffusion

In this chapter we state the PDE-form of the problem for this thesis, see Section 3.1.Then in Sections 3.2 and 3.3 we give notes on the physical interpretation of the modeland the available literature.

In Section 3.4 we will see that by rescaling the model some of its parameters canbe removed, making the notation shorter.

We conclude the chapter by presenting several choices for a nonlinear term in themodel and by proving some properties of this nonlinearity in Section 3.5.

3.1 The Cohen model

The model was proposed by Cohen and White in [CW91]. We state it here in amultidimensional version on a bounded domain, similar to [CWW95]. For simplicity, werefer to the problem as the Cohen model , although many more people have contributedto the formulation and understanding, see Section 3.3. Recall from Definition 2.1 and2.2 that Ω is an open, bounded domain with its boundary split into a Dirichlet partΓD and a Neumann part ΓN .

Problem 3.1 (The Cohen model).Find u = u(t,x), σ = σ(t,x) such that

u̇−∇TD∇u = f + ∇TK∇σ in Ω × [0, T ] (3.1a)σ̇ + γ(u)σ = µ u+ ν u̇+ f̂ in Ω × [0, T ] (3.1b)

u = uD on ΓD × [0, T ] (3.1c)(D∇u+K∇σ)T n = g on ΓN × [0, T ] (3.1d)

u = ŭ at t = 0 (3.1e)

σ = σ̆ at t = 0 (3.1f)

where D,K,µ > 0 are positive constants, ν a real constant and f = f(t,x), f̂ = f̂(t,x),g = g(t,x), uD = uD(t,x), ŭ = ŭ(x), and σ̆ = σ̆(x) are given functions.

The function γ in (3.1b) is a nonlinear function, for which we do not give a specificform. Instead we assume throughout this thesis that γ satisfies Assumption 3.4 givenbelow. The original model from [CWW95] has a specific form for γ that involves atanh-function, see Lemma 3.6.

Remark 3.2.The terms f and f̂ in Problem 3.1 are not part of the physical model, but introduced

33

CHAPTER 3. THE COHEN MODEL FOR NON-FICKIAN DIFFUSION 34

here to be able to prescribe exact solutions u, σ to the equations. This is important fortesting the implementation and showing convergence of the methods.

Remark 3.3 (The choice of ν = 0).The term νu̇ in (3.1b) was included in the model in [CW91] and in [HZ96]. However inboth articles, µ and ν are taken to be dependent on u. In [CWW95] however, ν = 0 waschosen. Because it would lead to more theoretical difficulties, we follow the example of[CWW95] and take ν = 0 for this thesis.

Assumption 3.4 (Properties of nonlinear function γ).Function γ : R → R must have these properties:

• γ is twice continuously differentiable: γ ∈ C2(R).

• There are constants γ̌ and γ̂ such that

0 < γ̌ ≤ γ(u) ≤ γ̂ ∀u ∈ R. (3.2)

• There is a constant C ′γ such that

0 ≤ γ′(u) ≤ C ′γ ∀u ∈ R. (3.3)

• There is a constant C ′′γ such that

|γ′′(u)| ≤ C ′′γ ∀u ∈ R. (3.4)

In Subsection 3.5.1 we will present three possible choices for γ that satisfy theseassumptions.

3.2 Physical interpretation of the model

Problem 3.1 is supposed to model the concentration u of certain substances inside apolymer. If this system could be described by the standard Fick’s law, then u wouldbe modelled by the heat equation:

u̇−D∆u = 0.

In order to explain experimental results, see [TW78], this equation was extended in[CW91]. Another term involving a new unknown function σ was added to get (3.1a).Because of similarities to the theory of viscoelastic materials, σ is often called ‘stress’.A second equation (3.1b) was added to describe how σ evolves with time. Note that(3.1b) can be solved explicitly for σ. With the simplifying assumptions that σ(0) = 0and ν = f̂ = 0 we get

σ =

∫ t

0exp(

−∫ t

sγ(u(ξ))dξ

)

µu(s) ds.

Inserting this in (3.1a) gives

u̇−∇TD∇u = f + ∇TK∇∫ t

0exp(

−∫ t

sγ(u(ξ))dξ

)

µu(s) ds.

It can be seen that the change in u depends on the whole history of u, i.e. all valuesu(s) for s ∈ (0, t). This is a property of viscoelastic materials, they possess a ‘memory’of their history.


The function γ in (3.1b) models a phase change of the polymer, depending on theconcentration u. If u is below a critical value ua, then the polymer is in a so-calledglassy state, which is characterised by a small constant value of γ. If, on the otherhand, u is above ua, then the polymer is in a so-called rubbery state and the value of γis also constant but considerably higher than in the glassy state, see [CW91]. In orderto avoid choosing a discontinuous jump function for γ, a small transition region aroundua was introduced and γ was taken to be constant (or almost constant) outside of thistransition region. This way γ can still be infinitely differentiable, see Subsection 3.5.1.

3.3 Notes on literature

The model was proposed by Cohen and White in [CW91], first deriving a general form,where the coefficients D, K, µ and ν are allowed to depend on u. This article alsointroduces the tanh-form of γ, given by Lemma 3.6. Later the special case is studiedwhere D, K, µ are constant and ν = 0.

This special case is also studied in [CWW95] on a two-dimensional bounded domain.In [Edw95], Edwards obtains solutions with a wave front advancing at constant

speed on the domain Ω = (0,∞). He replaces the tanh-form of γ by a discontinuousfunction taking only two values. This is generalised in [SE04] giving travelling wavesolutions. In that article D and K are dependent on u, µ and ν are constant and thereis a more general form for γ.

Rivière and Shaw have modified the problem in [RS06] (see also [RS05] for anextended version) and obtained a priori estimates for a discontinuous Galerkin finiteelement method applied to their modified problem. The equations they are studyingare

u̇−∇TD∇u−∇TKσ = f,σ̇ + γ(u)σ = µ∇u.

That is, the stress σ here is vector-valued and the first equation contains the divergenceof σ instead of the Laplacian of σ as in (3.1a). Moreover the second equation now hasthe gradient of u on the right-hand side. This modified problem appears to show similarbehaviour to Problem 3.1 but it is easier to prove estimates for it, see also the beginningof Chapter 6.

In [Vor08] Vorotnikov applies the technique of ‘dissipative solutions’ to the problemto obtain a certain type of weak solution in the whole space R2 or R3.

Amann shows in [Ama93] local existence and uniqueness for a generalisation of theproblem. For example, the factors D, K and µ are allowed to depend on the solutionand also the function γ is not assumed to be bounded. This paper applies very generaltechniques for quasilinear evolution equations to many different problems to show thepower of these methods.

A few years before, in [Ama91], Amann had studied the special problem that weare looking at and has shown global existence and uniqueness. This result is the mostrelevant for the form of equations that are studied in this thesis. Therefore we state itin Appendix B.

Hu and Zhang show in [HZ96] global existence and uniqueness for the generalisedproblem. They assume, instead of the term µu in the second equation a bounded termµ(u)u where µ(u) = 0 for any u which is greater than a fixed value. The method usedfor their proof is the Banach fixed point theorem.

In this thesis we will also look at the question of existence and uniqueness of solu-tions. Chapter 4 gives a uniqueness result in 1D for a variational formulation of the


problem.

3.4 Rescaling the equations

In order to simplify the problem without losing any of its properties, the variables canbe rescaled:

Lemma 3.5 (Rescaling the Cohen model).By scaling the time variable t, the space variable x and also σ in Problem 3.1, we canchoose D = K = µ = 1. Furthermore, suppose that ua is a special value of u (forexample, a Dirichlet boundary value or a certain value of function γ). By also scalingu, we can additionally get ua = 1.

Proof. Let A,B,M and E be arbitrary positive constants and introduce new scaledvariables:

ũ = Au, t̃ = Mt,

σ̃ = Bσ, x̃ = Ex.

Then

ut =MA ũt̃, ∆u =

E2

A ∆̃ũ,

σt =MB σ̃t̃, ∆σ =

E2

B ∆̃σ̃,

where the Laplacian for the scaled space variable is given by

∆̃ :=d∑

i=1

(

∂

∂x̃i

)2

.

Inserting these into (3.1a) and (3.1b), and then multiplying by AM (resp.BM ) gives

ũt̃ −DE2

M∆̃ũ− KE

2A

MB∆̃σ̃ =

A

Mf,

σ̃t̃ +γ(ũ/A)

Mσ̃ =

µB

AMũ+

ν B

Aũt̃ +

B

Mf̂.

As the coefficients do not contain a separate A but only the quotient B/A, we letH = B/A. Now, solving

DE2

M= 1,

KE2

MH= 1,

µH

M= 1,

for H, M and E, we obtain the following unique solution:

H =K

D, M =

µK

D, E =

√µK

D.

Note that, as the value of H is fixed by these, the coefficient containing ν cannot bescaled to ±1 as well.

It remains to choose a value for A and B. We choose A = 1/ua. Then u = uaimplies ũ = 1, as required. For the value of B we get B = AH = K/(Dua).

Thus after omitting the tilde marks from the new equations and redefining µ, f , f̂ ,g, γ, ŭ and σ̆, we get again the problem 3.1 but this time with D = K = µ = 1.


γ̂

γ̌

ua + 2∆auaua − 2∆a

γ(u

)

u

γ̂−γ̌2∆a

0


γ′ (u)

u

γ̂−γ̌2∆2a

0


γ′′ (u)

u

Figure 3.1: Graphs of γ = γ1 and its derivatives. This is the tanh-version of γ.

3.5 The nonlinearity

3.5.1 Choices for nonlinearity

In this subsection we present three possible choices for the nonlinear function γ.All these three functions have in common that they depend on constants γ̌ and γ̂

giving the minimum and maximum value and on two further constants ua and ∆a.The function value γ(u) is close to γ̌ for small values of u and close to γ̂ for large

values of u. In between there is a transition region around u = ua. The size of thistransition region is proportional to ∆a.

The first of the three choices for γ was introduced in [CW91] and was used in severalother articles, e.g. [CWW95], [RS06].

Lemma 3.6 (Usual choice for γ).Let 0 < γ̌ ≤ γ̂ and ua,∆a > 0. The specific γ = γ1, which is given by

γ1(u) =1

2(γ̂ + γ̌) +

1

2(γ̂ − γ̌) tanh

(

u− ua∆a

)

(3.5)

satisfies Assumption 3.4 with

C ′γ =γ̂ − γ̌2∆a

,

C ′′γ =γ̂ − γ̌∆a

2 (not optimal).

Proof. See [RS05].

See Figure 3.1 for a graph of this standard version of γ. The above choice of γhas the property that the transition region is ‘smeared out’, the values of γ̌ and γ̂ areattained only in the limit u → ±∞. For certain theoretical results, (see [NLM+]) it ismore useful to have a transition region, which is limited, and with γ constant outsideof this region. We now present two choices for function γ that have this property.


Lemma 3.7 (Another choice for γ).Let 0 < γ̌ ≤ γ̂ and ua,∆a > 0 and define

pγ(x) :=3

8x5 − 5

4x3 +

15

8x.

Then if function γ = γ2 is given by

γ2(u) :=

γ̌ if u ≤ ua − 12∆a,γ̂+γ̌

2 +γ̂−γ̌

2 pγ

(

u−ua12∆a

)

if |u− ua| < 12∆a,

γ̂ if u ≥ ua + 12∆a,

(3.6)

it satisfies Assumption 3.4 with

C ′γ =15(γ̂ − γ̌)

8∆a,

C ′′γ =10(γ̂ − γ̌)√

3∆a2.

Proof. The derivatives of pγ are

p′γ(x) =15

8x4 − 15

4x2 +

15

8=

15

8(x+ 1)2(x− 1)2,

p′′γ(x) =15

2x3 − 15

2x =

15

2(x+ 1)x(x− 1).

Hence we have that,

pγ(1) = 1, pγ(−1) = −1, p′γ(1) = p′γ(−1) = p′′γ(1) = p′′γ(−1) = 0,

which implies that γ ∈ C2(R). Furthermore, we see that for |u− ua| < 12∆a

γ′(u) =γ̂ − γ̌

2(12∆a)p′γ

(

u− ua12∆a

)

,

and

γ′′(u) =γ̂ − γ̌

2(12∆a)2p′′γ

(

u− ua12∆a

)

.

Since ‖p′γ‖L∞(−1,1) = 15/8 and ‖p′′γ‖L∞(−1,1) ≤ 15/2, we get the given size of theconstants.

See Figure 3.2 for a graph of this second version of γ.

Lemma 3.8 (A third choice for γ).Let 0 < γ̌ ≤ γ̂ and ua,∆a > 0 and define

f(x) := x+1

πsin(πx).

Then if function γ = γ3 is given by

γ3(u) :=

γ̌ if u ≤ ua − 12∆a,γ̂+γ̌

2 +γ̂−γ̌

2 f

(

u−ua12∆a

)

if |u− ua| < 12∆a,

γ̂ if u ≥ ua + 12∆a,

(3.7)


γ̂

γ̌

ua +12∆auaua − 12∆a

γ(u

)

u

15(γ̂−γ̌)8∆a

0


γ′ (u)

u

10(γ̂−γ̌)√3∆2a

0


γ′′ (u)

u

Figure 3.2: Graphs of γ = γ2 and its derivatives. This is the polynomial version of γ.

it satisfies Assumption 3.4 with

C ′γ =2(γ̂ − γ̌)

∆a,

C ′′γ =2π · (γ̂ − γ̌)

∆a2 .

Proof. The derivatives of f are

f ′(x) = 1 + cos(πx),

f ′′(x) = −π sin(πx).

Hence we have that,

f(1) = 1, f(−1) = −1, f ′(1) = f ′(−1) = f ′′(1) = f ′′(−1) = 0,

which implies that γ ∈ C2(R). Furthermore, we see that for |u− ua| < 12∆a

γ′(u) =γ̂ − γ̌

2(12∆a)f ′(

u− ua12∆a

)

,

and

γ′′(u) =γ̂ − γ̌

2(12∆a)2f ′′(

u− ua12∆a

)

.

Since ‖f ′‖L∞(−1,1) = 2 and ‖f ′′‖L∞(−1,1) = π, we get the given size of the constants.

See Figure 3.3 for a graph of this third version of γ.


γ̂

γ̌


γ(u

)

u

2(γ̂−γ̌)∆a

0


γ′ (u)

u

2π(γ̂−γ̌)∆2a

0


γ′′ (u)

u

Figure 3.3: Graphs of γ = γ3 and its derivatives. This is the sin(x) + x-version of γ.

3.5.2 Basic properties of γ

In this subsection two small lemmas involving differences of function values of γ aregiven. These will be needed later for proving the error estimate. Lemma 3.9 providesa bound for

‖γ(v) − γ(w)‖Lp(Ω),while Lemma 3.11 gives a bound for

‖γ(12v + 12w) − 12γ(v) − 12γ(w)‖L2(Ω).Lemma 3.9 (Difference of γ-values in Lp-norm).Let 1 ≤ p ≤ ∞ and v,w ∈ Lp(Ω). Then

‖γ(v) − γ(w)‖Lp(Ω) ≤ C ′γ‖v − w‖Lp(Ω).

Proof. From Taylor’s theorem we get that for all a, b ∈ R there is ϑ ∈ R with eithera ≤ ϑ ≤ b or b ≤ ϑ ≤ a such that

γ(b) − γ(a) = γ′(ϑ)(b− a).

Thus, for any a, b ∈ R,|γ(b) − γ(a)| ≤ C ′γ |b− a|.

This can be used together with the definition of the Lp-norm to finish the proof.

Before we state the second result of this subsection, Lemma 3.11, we prove thefollowing lemma:

Lemma 3.10 (A bound for C2-functions).If f : R → R is twice continuously differentiable with bounded second derivative,|f ′′(x)| ≤ α ∀x ∈ R, then

|f(12a+ 12b) − 12f(a) − 12f(b)| ≤α

8|b− a|2,

for all a, b ∈ R.


Proof. From Taylor’s theorem we get

f(a) = f(a+b2 ) +a−b2 f

′(a+b2 ) +12

(

a−b2

)2f ′′(�

Finite Element Methods for Non-Fickian Diffusion in Viscoelastic...

Documents

Transcript of Finite Element Methods for Non-Fickian Diffusion in Viscoelastic...