Optimization-based domain decomposition methods for multidisciplinary

simulation

by

Jeehyun Lee

A dissertation submitted to the graduate faculty

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Major: Applied Mathematics

Major Professor: Max D. Gunzburger

Iowa State University

Ames, Iowa

2001

Copyright c© Jeehyun Lee, 2001. All rights reserved.

ii

Graduate CollegeIowa State University

This is to certify that the Doctoral dissertation of

Jeehyun Lee

has met the dissertation requirements of Iowa State University

Committee Member

Committee Member

Committee Member

Committee Member

Committee Member

Major Professor

For the Major Program

For the Graduate College

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 FLUID-STRUCTURE INTERACTION PROBLEMS . . . . . . . . 5

2.1 The model problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 A weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The existence of a weak solution . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Finite element approximations . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 OPTIMIZATION-BASED DOMAIN DECOMPOSITION . . . . . 20

3.1 The model problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 The existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 21

3.3 Convergence with vanishing penalty parameter . . . . . . . . . . . . . . . 27

3.4 Optimality system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Sensitivity derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iv

ACKNOWLEDGMENTS

I first would like to thank Dr. Max D. Gunzburger for the expert guidance, almost

interminable patience that he has provided throughout this research and the writing of

this thesis. I am also grateful to my committee members for their efforts and contribu-

tions to this work : Dr. Qiang Du, Dr. L. Steven Hou, Dr. Justin Peters, Dr. Janet

S. Peterson and Dr. Monlay Tidriri. Special thanks to Dr. Xioaming Wang who was

generous to share his knowledge and to answer any questions about partial differential

equations. Not a day of my life passes by without thoughts and fond memories of my

family. A big thank to my parents, parents-in-law and my sister, Jeena for their encour-

agement and for believing in me. I want to express my deepest thanks to my husband,

Wook for his love, friendship and unfailing support throughout my career in graduate

school and to my lovely boy, Daniel who is the joy of my life.

v

ABSTRACT

We consider a domain decomposition method for a fluid-structure interaction prob-

lem. The fluid-structure interaction problem involves two mathematical models, each

posed on a different domain, so that domain decomposition occurs naturally. Our ap-

proach to a domain decomposition method is based on a strategy in which unknown data

at the interface is determined through an optimization process. We prove that the solu-

tion of the optimization problem exists. And we show that the Lagrange multiplier rule

may be used to transform the constrained optimization problem into an unconstrained

one and that rule is applied to derive an optimality system from which optimal solutions

may be obtained. We then study a gradient method for solving optimization problem.

1

1 INTRODUCTION

Multidisciplinary simulation problems arise in a variety of settings in which more

than one media, or more than one mathematical model, or more than one dominant

effect are present. The direct solution of such problems presents a formidable challenge,

since they usually involve large, coupled system of partial differential equations. For

this reason, methods which uncouple the different disciplines are of interest. Here, we

discuss uncoupling procedures which are based on using an optimization strategy.

A main virtue of our approach is that it allows for the user to use existing codes for

each discipline as black boxes and only requires that the user write a simple code that

effects the coupling between the disciplines. One reason we are able to do this is that

our methodology allows for complete flexibility with regards to the boundary conditions

imposed on each discipline. Another virtue of the optimizaion-based decoupling is that

it allows for the use of efficient iterative strategies, e.g., quickly converging iterative pro-

cesses by which solutions of the coupled, multidisciplinary problem are determined. Our

methodology has other important virtues as well as allowing for the use of mismatched

grids and different discretization methods for each discipline.

Here, for the sake of concreteness, we will describe the optimization-based domain

decomposition method for a fluid-structure interaction problem. The subjects of fluid-

structure interaction problem have been extensively studied in the past and continue

to be the focus of much attention today. There are number of different types of math-

ematical models for fluid-structure interactions. We classify these models into three

categories.

2

Elementary fluid. The fluid motion is governed by potential equations, e.g., Laplace

equations or wave equations. In [34], a coupled system of a potential equation and a

wave equation is considered. Elementary fluids coupled with rigid cavity or moving wall

has been studied in [19] and with an elastic solid in [4].

Inviscid fluid. A few mathematical papers have appeared for fluid-structure interac-

tions modeled using inviscid fluid models, e.g., the Euler equations. Interactions between

linearized inviscid fluids and elastic solids have been analyzed in [2],[35]. An algorithm

for an inviscid nonlinear fluid coupled with rigid walls was given in [3].

Viscous fluid. There is an extensive literature on linearized viscous fluids coupled

with solids. Solids modeled using plate equations or shell equations are treated in

[15],[16],[17],[32]. The Stokes equations coupled with a beam equation has been ana-

lyzed in [21]. In [11], [31], interactions between linearized viscous fluid and elastic solids

are studied. See [7] interaction with rigid walls.

Also, there is a vast literature on fluid-structure interactions for which the fluid is

modeled using nonlinear viscous fluid models, e.g., the Navier-Stokes equations. Rigid

body motions of solids in a nonlinear viscous fluid have been studied in [8],[12], [22], [23].

In [18], a coupled problem of Navier-Stokes equations and a plate equation is studied.

Some works have treated interactions between nonlinear viscous fluids and elastic solids.

See, e.g., [8],[13],[14],[36],[37].

MODEL PROBLEM

In this paper, we consider elastic body motions in a fluid flow. Let Ωf and Ωs denote

the regions occupied by the fluid and solid, respectively. Let Γ0 denote the interface

between the fluid and solid and let Γf and Γs denote the boundaries of the fluid and

solid regions (other than the interface Γ0). In the fluid region, we apply the Stokes

System.

3

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

v|t=0 = v0 in Ωf

(1.1)

Here, ρf and µf denote the (constant) fluid density and viscosity, v the fluid velocity,

p the fluid pressure, and v0 the initial velocity.

In the solid, we apply the equation of the linear elasticity.

ρsutt − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

(1.2)

Here, µf and λf are the Lame constants and ρs the constant density of solid, b

denotes a given loading force per unit mass, u the displacement of the solid, and u0 and

u1 are given initial data.

Along the fixed interface Γ0 between the fluid and solid, the velocity of the fluid and

solid are equal, as are the stress vector in the fluid and solid. Thus, we have

ut = v on Γ0 (1.3)

and

µs∇u · n + (λs + µs)(∇ · u)n = pn− µf∇v · n on Γ0 (1.4)

Solving (1.1)-(1.4) is a formidable challenge. Fluid-structure interaction problems

involve two different mathematical models, each posed on a different domain, so that

domain decomposition occurs naturally. Our approach to domain decomposition is based

4

on a strategy in which unknown data at the interface is determined through an opti-

mization process. We consider the following interface conditions.

v = g on Γ0 (1.5)

and

µs∇u · n + (λs + µs)(∇ · u)n = h on Γ0 (1.6)

Then we may solve (1.1) and (1.5) for v and p and solve (1.2) and (1.6) for u.

For an arbitrary choice of g and h, (1.1) and (1.2) are satisfied. However, (1.3) and

(1.4) will not be satisfied. On the other hand, we know that g and h exist such that

solutions of (1.1),(1.5) and (1.2),(1.6) are solutions of (1.1)-(1.4). We merely have to

choose g = v|Γ0 = ut|Γ0 and h = µs∇u · n + (λs + µs)(∇ · u)n = pn− µf∇v · n on Γ0,

where (v, p, u) is a solution of (1.1)-(1.4).

The optimization-based domain decomposition algorithm finds such g and h by min-

imizing the functional

J (v, p,g,u,h) =1

2

∫ T

0

∫Γ0

(ut − g)2 dΓ dt

+1

2

∫ T

0

∫Γ0

(pn− µf∇v · n− h)2dΓ dt

In the remainder of this chapter, we discuss the derivation of the model (1.1)-(1.4).

This thesis is organized as follows. In chapter 2, a coupled system of fluid-structure

interaction problem is studied. The existence of a weak solution is proved and finite ele-

ment approximations are discussed. In chapter 3, we introduce an optimization problem

to uncouple the system. We prove that the optimization problem has a solution and the

solution converges to the solution of the coupled system. The Lagrange multiplier rule

is used to derive an optimality system from which optimal solutions may be determined.

Finally, we define a gradient method for the solution of the optimality system and show

the convergence for the gradient method.

5

2 FLUID-STRUCTURE INTERACTION PROBLEMS

2.1 The model problems

We consider a coupled system of Stokes and elasticity problem that was introduced

in the previous chapter.

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

ρsutt − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

ut = v on Γ0

µs∇u · n + (λs + µs)(∇ · u)n = pn− µf∇v · n on Γ0

v|t=0 = v0 in Ωf

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

(2.1.1)

2.2 Notation

Throughout this paper, C will denote a positive constant whose meaning and value

changes with context. Hs(D), s ∈ IR, denotes the standard sobolev space of order s

6

with respect to the set D, equipped with the standard norm ‖ · ‖s,D. Corresponding

sobolev spaces of vector-valued functions will be denoted by Hs(D). Dual spaces will

be denoted by (.)∗. We then define the subspaces

H1f (Ωf ) = v ∈ H1(Ωf ) : v = 0 on Γf

J = v ∈ D(Ωf ) : ∇ · v = 0 on Ωf

where D(Ωf ) be the space of C∞ functions with compact support contained in Ωf .

V = the closure of J in H1f (Ωf )

H = the closure of J in L2(Ωf )

and

H1s(Ωs) = u ∈ H1(Ωs) : u = 0 on Γs

We define, for (pq) ∈ L1(D) and (u · v) ∈ L1(D),

(p, q)D =∫

Dpq dD and (u,v)D =

∫D

u · v dD

respectively.

We define the bilinear forms

af (u,v) =∫Ωf

µf∇u : ∇v dΩ ∀u,v ∈ H1(Ωf )

as(u,v) =∫Ωs

µs∇u : ∇v + (λs + µs)(∇ · u)(∇ · v) dΩ ∀u,v ∈ H1(Ωs)

a(u,v) =∫Ωs

µs∇u : ∇v dΩ ∀u,v ∈ H1(Ωs)

and

b(v, q) = −∫Ωf

q divv dΩ ∀v ∈ H1(Ωf ), ∀q ∈ L2(Ωf )

It is well known that the forms af (·, ·), as(·, ·), a(·, ·) and b(·, ·) are continuous, i.e.

there exist positive constants ka and kb such that

|af (u,v)| ≤ ka‖u‖1,Ωf‖v‖1,Ωf

∀u,v ∈ H1(Ωf )

7

|as(u,v)| ≤ ka‖u‖1,Ωs‖v‖1,Ωs ∀u,v ∈ H1(Ωs)

|a(u,v)| ≤ ka‖u‖1,Ωs‖v‖1,Ωs ∀u,v ∈ H1(Ωs)

and

|b(v, q)| ≤ kb‖v‖1,Ωf‖q‖0,Ωf

∀v ∈ H1(Ωf ), ∀q ∈ L2(Ωf )

Also, af (·, ·), as(·, ·) and a(·, ·) satisfy the coercivity property and b(·, ·) satisfies the

inf-sup condition, which means there exist positive constants Ka and Kb such that

|af (u,u)| ≥ Ka‖u‖21,Ωf

∀u ∈ H1(Ωf )

|as(u,u)| ≥ Ka‖u‖21,Ωs

∀u ∈ H1(Ωf )

|a(u,u)| ≥ Ka‖u‖21,Ωs

∀u ∈ H1(Ωf )

and

inf0 6=q∈L2(Ωf )

sup0 6=v∈H1

0(Ωf )

b(v, q)

‖v‖1,Ωf‖q‖0,Ωf

≥ Kb

2.3 A weak formulation

We define a space of trial functions and test functions as

U = (v,u) : v ∈ L2(0, T ; H1f (Ωf )), u ∈ L2(0, T ; H1

s(Ωs)),

ut ∈ L2(0, T ; L2(Ωs)), such that v = ut on Γ0

A weak formulation corresponding to (2.1.1) is given by

For f ,v0,b,u0 and u1 given,

f ∈ L2(0, T ; H1f (Ωf )∗) (2.3.2)

v0 ∈ L2(Ωf ) (2.3.3)

b ∈ L2(0, T ; L2(Ωs)) (2.3.4)

u0 ∈ H1s(Ωs) (2.3.5)

u1 ∈ L2(Ωs) (2.3.6)

8

to find (v,u) ∈ U satisfying

ρf (vt,w)Ωf+ b(w, p) + af (v,w) + ρs(utt, θt)Ωs

+ as(u, θt) = ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ U (2.3.7)

b(v, q) = 0 ∀q ∈ L2(Ωf ) (2.3.8)

v|t=0 = v0 (2.3.9)

u|t=0 = u0 (2.3.10)

ut|t=0 = u1 (2.3.11)

We recall the definition of the spaces J, V and H, introduced in the previous section

and which will be the basic spaces for W .

J = v ∈ D(Ωf ) : ∇ · v = 0 on Ωf

V = the closure of J in H1f (Ωf )

H = the closure of J in L2(Ωf )

Now, we define a space of trial functions and test functions as

W = (v,u) : v ∈ L2(0, T ;V ), u ∈ L2(0, T ; H1s(Ωs)),

ut ∈ L2(0, T ; L2(Ωs)), such that v = ut on Γ0

Then a weak formulation (2.3.7)-(2.3.11) is equivalent to the following:

To find (v,u) ∈ W satisfying

ρf (vt,w)Ωf+ af (v,w) + ρs(utt, θt)Ωs + as(u, θt)

= ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ W (2.3.12)

v|t=0 = v0 (2.3.13)

u|t=0 = u0 (2.3.14)

ut|t=0 = u1 (2.3.15)

9

2.4 The existence of a weak solution

To show the existence of weak solutions for the coupled system, we introduce an

auxiliary problem involving an ’artificial viscosity’ term. The auxiliary problem is defined

as follows.

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

ρsutt − εµs∆ut − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

ut = v on Γ0

εµs∇ut · n + µs∇u · n + (λs + µs)(∇ · u)n = pn− µf∇v · n on Γ0

v|t=0 = v0 in Ωf

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

(2.4.16)

We define a space of trial functions and test functions for the auxiliary problem as

U = (v,u) : v ∈ L2(0, T ; H1f (Ωf )), u ∈ L2(0, T ; H1

s(Ωs)),

ut ∈ L2(0, T ; H1s(Ωs)), such that v = ut on Γ0

A weak formulation corresponding to (2.4.16) is given by

For f ,v0,b,u0 and u1 given,

f ∈ L2(0, T ; H1f (Ωf )∗) (2.4.17)

v0 ∈ L2(Ωf ) (2.4.18)

10

b ∈ L2(0, T ; L2(Ωs)) (2.4.19)

u0 ∈ H1s(Ωs) (2.4.20)

u1 ∈ L2(Ωs) (2.4.21)

to find (v,u) ∈ U satisfying

ρf (vt,w)Ωf+ b(w, p) + af (v,w) + ρs(utt, θt)Ωs + εa(ut, θt)

+ as(u, θt) = ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ U (2.4.22)

b(v, q) = 0 ∀q ∈ L2(Ωf ) (2.4.23)

v|t=0 = v0 (2.4.24)

u|t=0 = u0 (2.4.25)

ut|t=0 = u1 (2.4.26)

Now, we define a space of trial functions and test functions as

W = (v,u) : v ∈ L2(0, T ;V ), u ∈ L2(0, T ; H1s(Ωs)),

ut ∈ L2(0, T ; H1s(Ωs)), such that v = ut on Γ0

Then a weak formulation (2.4.22)-(2.4.26) is equivalent to the following:

To find (v,u) ∈ W satisfying

ρf (vt,w)Ωf+ af (v,w) + ρs(utt, θt)Ωs + εa(ut, θt)

+ as(u, θt) = ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ W (2.4.27)

v|t=0 = v0 (2.4.28)

u|t=0 = u0 (2.4.29)

ut|t=0 = u1 (2.4.30)

We show the existence of solutions of the auxiliary problem in the next theorem.

11

Theorem 2.4.1 For given f ,v0,b,u0 and u1 which satisfy (2.4.17)-(2.4.21), there ex-

ists a unique solution (v,u) ∈ W which satisfy (2.4.27)-(2.4.30). Moreover v ∈ L∞(0, T ;H), u ∈

L∞(0, T ; H1s(Ωs)) and ut ∈ L∞(0, T ; L2(Ωs))

proof: We use the Galerkin method. Let (φn)n∈N be a basis of V and (θn)n∈N be a

basis of H1s(Ωs) such that φn = θn on Γ0. We define discrete spaces Mn = spanφm, 1 ≤

m ≤ n and Nn = spanθm, 1 ≤ m ≤ n. Also define a discrete space of trial and test

functions by

Vn = (v,u) ∈ Vn,f × Vn,s, ut = v on Γ0

with

Vn,f = v =n∑

i=1

ai(t)φi, ai(t) ∈ H1(0, T )

Vn,s = u =n∑

i=1

bi(t)θi, bi(t) ∈ H1(0, T )

The discrete problem is : Find (vn,un) ∈ Vn such that vn(0) ∈ Mn, u(0) ∈ Nn and

unt(0) ∈ Nn with vn(0) → v0 in L2(Ωf ), un(0) → u0 in H1(Ωs) and unt(0) → u1 in

L2(Ωs) and

ρf (vnt,wn)Ωf+ af (vn,wn) + ρs(untt, znt)Ωs + εa(unt, znt)

+as(un, znt) = ρf (f ,wn)Ωf+ ρs(b, znt)Ωs ∀(wn, zn) ∈ Vn

(2.4.31)

Since this is a linear system of ordinary differential equations, there is a unique solution.

We will obtain a priori estimates independent of n for the functions vn,un and then

pass to the limit. Set wn = vn and zn = un in (2.4.31). We get

ρf (vnt,vn)Ωf+ af (vn,vn) + ρs(untt,unt)Ωs + εa(unt,unt)

+as(un,unt) = ρf (f ,vn)Ωf+ ρs(b,unt)Ωs

(2.4.32)

and this gives

ρf

2

d

dt‖vn‖2

0,Ωf+Ka‖vn‖2

1,Ωf+ρs

2

d

dt‖unt‖2

0,Ωs+ εKa‖unt‖2

1,Ωs

+Ka

2

d

dt‖un‖2

1,Ωs≤ ρf‖f‖−1,Ωf

‖vn‖1,Ωf+ ρs‖b‖0,Ωs‖unt‖0,Ωs

12

Integrating this from 0 to t, we obtain

ρf

2(‖vn‖2

0,Ωf− ‖vn(0)‖2

0,Ωf) +Ka

∫ t

0‖vn‖2

1,Ωf+ρs

2(‖unt‖2

0,Ωs− ‖unt(0)‖2

0,Ωs)

+ εKa

∫ t

0‖unt‖2

1,Ωs+Ka

2(‖un‖2

1,Ωs− ‖un(0)‖2

1,Ωs)

≤ρ2

f

2Ka

∫ t

0‖f‖2

−1,Ωf+Ka

2

∫ t

0‖vn‖2

1,Ωf+ Tρs

∫ t

0‖b‖2

0,Ωs+ρs

2T

∫ t

0‖unt‖2

0,Ωs

Gronwall’s inequality may be used to conclude

suptρf‖vn‖2

0,Ωf+Ka

∫ T

0‖vn‖2

1,Ωf+ sup

tρs‖unt‖2

0,Ωs+ εKa

∫ T

0‖unt‖2

1,Ωs

+ suptKa‖un‖2

1,Ωs≤ C

∫ T

0‖f‖2

−1,Ωf+ ρsC

∫ T

0‖b‖2

0,Ωs

+ ρf‖v0‖20,Ωf

+ ρs‖u1‖20,Ωs

+Ka‖u0‖21,Ωs

Hence vn is uniformly bounded in L∞(0, T ;H) and L2(0, T ;V ); un is uni-

formly bounded in L∞(0, T ; H1s(Ωs)); unt is uniformly bounded in L∞(0, T ; L2(Ωs))

and L2(0, T ; H1s(Ωs)); Thus, there exist weakly convergent subsequences and by passing

to the limit, a solution of (2.4.27)-(2.4.30) exists.

Uniqueness. Let (v1,u1) and (v2,u2) be two solutions of (2.4.27)-(2.4.30). Then

energy estimates may be used to get

suptρf‖v1 − v2‖2

0,Ωf+Ka

∫ T

0‖v1 − v2‖2

1,Ωf+ sup

tρs‖u1t − u2t‖2

0,Ωs

+ εKa

∫ T

0‖u1t − u2t‖2

1,Ωs+ sup

tKa‖u1 − u2‖2

1,Ωs≤ 0

and hence v1 = v2 and u1 = u2.

Now we show the existence of the coupled system by taking the limit of solutions of

the auxiliary problem as ε→ 0

Theorem 2.4.2 For given f ,v0,b,u0 and u1 which satisfy (2.3.2)-(2.3.6), there exists a

unique solution (v,u) ∈ W which satisfy (2.3.12)-(2.3.15). Moreover v ∈ L∞(0, T ;H), u ∈

L∞(0, T ; H1s(Ωs)) and ut ∈ L∞(0, T ; L2(Ωs))

13

proof: ∀ε, there exists vε,uε which are solutions of (2.4.27)-(2.4.30). Using a priori

estimate obtained in the proof of Theorem 2.4.1,

suptρf‖vε‖2

0,Ωf+Ka

∫ T

0‖vε‖2

1,Ωf+ sup

tρs‖(uε)t‖2

0,Ωs

+ εKa

∫ t

0‖(uε)t‖2

1,Ωs+ sup

tKa‖uε‖2

1,Ωs≤ C(v0,u0,u1, f ,b)

From which we deduce

vε is uniformly bounded in L2(0, T ;V ) ∩ L∞(0, T ;H)

uε is uniformly bounded in L∞(0, T ; H1s(Ωs))

(uε)t is uniformly bounded in L∞(0, T ; L2(Ωs))

√ε(uε)t is uniformly bounded in L2(0, T ; H1

s(Ωs))

Therefore, we can extract a subsequence such that

vε v in L2(0, T ;V ) ∩ L∞(0, T ;H)

uε u in L∞(0, T ; H1s(Ωs))

(uε)t ut in L∞(0, T ; L2(Ωs))

Since√ε(uε)t is uniformly bounded in L2(0, T ; H1

s(Ωs)) then√ε(∆uε)t is uniformly

bounded in L2(0, T ; H1s(Ωs)

∗) and

limε→0

εas((uε)t,w) = 0 ∀w ∈ L2(0, T ; H1s(Ωs))

From (2.4.27),

ρf ((vε)t,w)Ωf+ ρs((uε)tt, θt)Ωs

= −af (vε,w)− εa((uε)t, θt)− as(uε, θt) + ρf (f ,w)Ωf+ ρs(b, θt)Ωs

→ −af (v,w)− as(u, θt) + ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ W

Moreover, since vε v in L∞(0, T ;H) and (uε)t ut in L∞(0, T ; L2(Ωs)) we have

ρf (vt,w)Ωf+ ρs(utt, θt)Ωs

= −af (v,w)− as(u, θt) + ρf (f ,w)Ωf+ ρs(b, θt)Ωs ∀(w, θ) ∈ W

14

Further,

vε|t=0 v|t=0 in H

uε|t=0 u|t=0 in H1s(Ωs)

uεt|t=0 ut|t=0 in L2(Ωs)

Thus the existence theorem is proved. The proof of uniqueness is exactly the same

as before.

2.5 Finite element approximations

Let h denote a discretization parameter tending to zero and, for each h, let Xhf ,

Sh and Xhs be finite dimensional spaces such that Xh

f ⊂ H1f (Ωf ), Sh ⊂ L2(Ωf ) and

Xhs ⊂ H1

s (Ωs). And

Uh = (vh,uh) : vh ∈ L2(0, T ; Xhf ), uh ∈ H1(0, T ; Xh

s ),

such that vh = uht on Γ0

We assume that the finite element spaces satisfy the standard approximation prop-

erties, i.e.,

infv∈Xh

f

‖v − vh‖1,Ωf≤ ‖v‖m+1,Ωf

∀v ∈ Hm+1(Ωf )

infp∈Sh

‖p− ph‖0,Ωf≤ ‖p‖m,Ωf

∀p ∈ Hm(Ωf )

and

infu∈Xh

s

‖u− uh‖1,Ωs ≤ ‖u‖m+1,Ωs ∀u ∈ Hm+1(Ωs)

We also assume the inf-sup condition (or LBB) condition,

inf0 6=qh∈Sh

sup0 6=vh∈Xh

f

b(vh, qh)

‖vh‖1,Ωf‖qh‖0,Ωf

≥ Kb

where Kb is a positive constant independent of h.

15

Finite element approximations of solutions of the coupled system (2.3.7)-(2.3.11) are

defined as follows: Seek (vh,uh, ph) ∈ Uh × L2(0, T ;Sh) such that

ρf (vht ,w

h)Ωf+ b(wh, ph) + af (vh,wh) + ρs(u

htt, θ

ht )Ωs + as(u

h, θht )

= ρf (f ,wh)Ωf+ ρs(b, θ

ht )Ωs ∀(wh, θh) ∈ Uh (2.5.33)

b(vh, qh) = 0 ∀qh ∈ L2(0, T ;Sh) (2.5.34)

vh|t=0 = v0h (2.5.35)

uh|t=0 = u0h (2.5.36)

uht |t=0 = u1h (2.5.37)

First, we show the convergence of finite element approximations. To show the con-

vergence, we consider finite element approximations of the auxiliary problem (2.4.22)-

(2.4.26): Seek (vεh,uεh, pεh) ∈ Uh × L2(0, T ;Sh) such that

ρf (vεht ,w

h)Ωf+ b(wh, pεh) + af (vεh,wh) + ρs(u

εhtt , θ

ht )Ωs

+ εa(uεht , θ

ht ) + as(u

εh, θht )

= ρf (f ,wh)Ωf+ ρs(b, θ

ht )Ωs ∀(wh, θh) ∈ Uh (2.5.38)

b(vεh, qh) = 0 ∀qh ∈ L2(0, T ;Sh) (2.5.39)

vεh|t=0 = v0h (2.5.40)

uεh|t=0 = u0h (2.5.41)

uεht |t=0 = u1h (2.5.42)

Lemma 2.5.1 For each ε ≥ 0, let (vε,uε, pε) denote solutions of auxiliary problem

(2.4.22)-(2.4.26) and (vεh,uεh, pεh) denote finite element approximations of solutions of

auxiliary problem. Then (vεh,uεh, pεh) → (vε,uε, pε) in L2(0, T ; H1(Ωf ))×H1(0, T ; H1(Ωs))×

H−1(0, T ;L2(Ωf )) as h→ 0.

proof: Set wh = vεh and θh = uεh in (2.5.38) and combine with (2.5.39), we obtain

suptρf‖vεh‖2

0,Ωf+Ka

∫ T

0‖vεh‖2

1,Ωf+ sup

tρs‖uεh

t ‖20,Ωs

16

+ εKa

∫ t

0‖uεh

t ‖21,Ωs

+ suptKa‖uεh‖2

1,Ωs

≤ C∫ T

0‖f‖2

−1,Ωf+ ρsC

∫ T

0‖b‖2

0,Ωs

+ ρf‖v0‖20,Ωf

+ ρs‖u1‖20,Ωs

+Ka‖u0‖21,Ωs

The same argument as in the proof of the Theorem 2.4.1 yields (vεh,uεh) → (vε,uε)

in L2(0, T ; H1(Ωf ))×H1(0, T ; H1(Ωs)) as h→ 0. If we take pεh satisfying (2.5.38) and pε

satisfying (2.4.22) then pεh → pε in H−1(0, T ;L2(Ωf )) as h→ 0. Thus lemma is proved.

It was proved in Theorem 2.4.2 that (vε,uε, pε) → (v,u, p) as ε → 0 and it can be

shown (vεh,uεh, pεh) → (vh,uh, ph) as ε → 0 in the same manner. Combining this with

Lemma 2.5.1 gives the following theorem.

Theorem 2.5.2 Let (v,u, p) denote solutions of the coupled system (2.3.7)-(2.3.11)

and (vh,uh, ph) denote solutions of (2.5.33)-(2.5.37). Then (vh,uh, ph) → (v,u, p) in

L2(0, T ; H1(Ωf ))× (L2(0, T ; H1(Ωs) ∩H1(0, T ; L2(Ωs))×H−1(0, T ;L2(Ωf )) as h→ 0.

2.6 Error estimates

For the purpose of the proof of next theorem, we introduce some spaces and a pro-

jection. Let Ω denote Ωf ∩ Ωs and Xh denote a finite dimensional space such that

Xh ∈ H1(Ω). We define a continous space G and a discrete space Gh by

G = z ∈ H1(Ω) : zf ≡ z|Ωf∈ V, zs ≡ z|Ωs ∈ H1(Ωs), and zf |Γ0 = zs|Γ0

Gh = z ∈ Xh : ψhf ≡ ψh|Ωf

∈ Xhf , ψ

hs ≡ ψh|Ωs ∈ Xh

s ,

b(ψhf , q

h) = 0 ∀qh ∈ Sh, and ψhf |Γ0 = ψh

s |Γ0

We define P h : G −→ Gh to be the projection with respect to the L2(Ω) inner

product, i.e. P hz = z if

(z, ψh)Ω = (z, ψh)Ω ∀ψ ∈ Gh, ∀z ∈ G

17

Then

‖z− z‖1,Ω ≤ Chm‖z‖m+1,Ω (2.6.43)

‖z− z‖0,Ω ≤ Chm+1‖z‖m+1,Ω (2.6.44)

Now we prove the following estimate for the error between the solutions of the semi-

discrete and continuous problems.

Let

ζ =

v on Ωf

ut on Ωs

And define v = P hζ|Ωf, ut = P hζ|Ωs ,

Theorem 2.6.1

ρf‖v − vh‖20,Ωf

+Ka

∫ t

0‖v − vh‖2

1,Ωfdt

+ρs‖ut − uht ‖2

0,Ωs+Ka‖u− uh‖2

1,Ωs

≤ Ch2m+2‖v‖2m+1,Ωf

+ Ch2m∫ t

0‖v‖2

m+1,Ωfdt

+Ch2m‖u‖2m+1,Ωs

+ Ch2m+2‖ut‖2m+1,Ωs

+Ch2m∫ t

0‖u‖2

m+1,Ωsdt+ Ch2m

∫ t

0‖p‖2

m,Ωfdt (2.6.45)

proof: By subtracting (2.5.33)-(2.5.34) from the corresponding equations of (2.3.7)-

(2.3.8) we obtain the “orthogonality conditions”.

ρf (vt − vht ,w

h)Ωf+ b(wh, p− ph) + af (v − vh,wh)

+ ρs(utt − uhtt, θ

ht )Ωs + as(u− uh, θh

t ) = 0 ∀(wh, θh) ∈ Uh (2.6.46)

b(v − vh, qh) = 0 ∀qh ∈ L2(0, T ;Sh) (2.6.47)

Using (2.6.46)-(2.6.47) we deduce that

ρf (vt − vht ,v − vh)Ωf

+ af (v − vh,v − vh)

18

+ρs(utt − uhtt,ut − uh

t )Ωs + as(u− uh,ut − uht )

= ρf (vt − vht ,v − v)Ωf

+ af (v − vh,v − v) + ρs(utt − uhtt,ut − ut)Ωs

+as(u− uh,ut − ut)− b(v − vh, p− ph)

+ρf (vt − vht , v − vh)Ωf

+ af (v − vh, v − vh) + ρs(utt − uhtt, ut − uh

t )Ωs

+as(u− uh, ut − uht ) + b(v − vh, p− ph)

= ρf (vt − vht ,v − v)Ωf

+ af (v − vh,v − v) + ρs(utt − uhtt,ut − ut)Ωs

+as(u− uh,ut − ut)− b(v − vh, p− ph) (2.6.48)

Then using the facts that (ψhf ,v − v) = 0 ∀ψh

f ∈ Xhf and that

b(v, qh) = 0 = b(vh, qh) ∀qh ∈ Sh (2.6.49)

We obtain (by also noting that (2.6.49) implies vt ∈ Xhf and vh

t ∈ Xhf )

ρf (vt − vht ,v − v)Ωf

= ρf (vt,v − v)Ωf= ρf (vt − vt,v − v)Ωf

(2.6.50)

Similarly,

ρs(utt − uhtt,ut − ut)Ωs = ρs(utt − utt,ut − ut)Ωs (2.6.51)

Combining (2.6.48)-(2.6.51) we deduce that for all qh ∈ L2(0, T ;Sh)

ρf

2

d

dt‖v − vh‖2

0,Ωf+Ka‖v − vh‖2

1,Ωf

+ρs

2

d

dt‖ut − uh

t ‖20,Ωs

+Ka

2

d

dt‖u− uh‖2

1,Ωs

≤ ρf

2

d

dt‖v − v‖2

0,Ωf+ ka‖v − vh‖1,Ωf

‖v − v‖1,Ωf

+ρs

2

d

dt‖ut − ut‖2

0,Ωs+ ka‖u− uh‖1,Ωs‖ut − ut‖1,Ωs

+kb‖v − v‖1,Ωf‖p− qh‖0,Ωf

+ kb‖v − vh‖1,Ωf‖p− qh‖0,Ωf

Therefore,

ρf‖v − vh‖20,Ωf

+Ka

∫ t

0‖v − vh‖2

1,Ωfdt+ ρs‖ut − uh

t ‖20,Ωs

+Ka‖u− uh‖21,Ωs

≤ C(‖v − v‖20,Ωf

+∫ t

0‖v − v‖2

1,Ωfdt+ ‖u(t1)− u(t1)‖2

1,Ωs

+‖ut − ut‖20,Ωs

+∫ t

0‖ut − ut‖2

1,Ωsdt+

∫ t

0‖p− qh‖2

0,Ωfdt) (2.6.52)

19

for all qh ∈ L2(0, T ;Sh), where ‖u(t1)− u(t1)‖ = maxt∈[0,T ] ‖u(t)− u(t)‖. Hence (2.6.45)

follows from (2.6.52), (2.6.43) and (2.6.44).

20

3 OPTIMIZATION-BASED DOMAIN DECOMPOSITION

In this chapter, we present an optimization-based domain decomposition method to

uncouple the computations.

3.1 The model problems

Let v, p,g,u and h satisfy the coupled system (2.1.1). Instead of constraints (2.1.1),

we consider the uncoupled system

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

v = g on Γ0

ρsutt − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

µs∇u · n + (λs + µs)(∇ · u)n = h on Γ0

v|t=0 = v0 in Ωf

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

(3.1.1)

In this paper, we refer to g and h as controls. Our goal is to find g and h such

21

that the solutions of (3.1.1) coincide with solutions of (2.1.1). The optimization-based

domain decomposition algorithm finds such g and h by minimizing the functional

J (v, p,g,u,h) =1

2

∫ T

0

∫Γ0

(ut − g)2 dΓ dt

+1

2

∫ T

0

∫Γ0

(pn− µf∇v · n− h)2dΓ dt

In order to regularize the optimization problem, we instead minimize the penalized

functional

J δ(v, p,g,u,h) =1

2

∫ T

0

∫Γ0

(ut − g)2 dΓ dt

+1

2

∫ T

0

∫Γ0

(pn− µf∇v · n− h)2dΓ dt+δ

2

∫ T

0

∫Γ0

g2 dΓ dt

+δ

2

∫ T

0

∫Γ0

(∇Γg)2 dΓ dt+δ

2

∫ T

0

∫Γ0

g2t dΓ dt+

δ

2

∫ T

0

∫Γ0

h2 dΓ dt

where the penalty parameter δ is a positive constant that can be chosen to change the

relative importance of penalty terms in Jδ and ∇Γ denotes tangential gradient. Thus

the optimization problem we propose to solve is given by

Problem 1

Find (v,p,g,u,h) such that the functional Jδ is minimized subject to (3.1.1).

3.2 The existence of an optimal solution

We recall the definition of the spaces

H1f (Ωf ) = v ∈ H1(Ωf ) : v = 0 on Γf

J = v ∈ D(Ωf ) : ∇ · v = 0 on Ωf

V = the closure of J in H1f (Ωf )

H = the closure of J in L2(Ωf )

and

H1s(Ωs) = u ∈ H1(Ωs) : u = 0 on Γs

22

For functions defined on Γ0, we will use the subspaces

Y = g ∈ L2(0, T ; H1(Γ0)) ∩H1(0, T ; L2(Γ0)) : g = 0 on ∂Γ0

Z = L2(0, T ; L2(Γ0))

with norms

‖g‖2Y =

∫ T

0

∫Γ0

g2 dΓ dt+∫ T

0

∫Γ0

(∇Γg)2 dΓ dt+∫ T

0

∫Γ0

g2t dΓ dt

‖h‖2Z =

∫ T

0

∫Γ0

h2 dΓ dt

A weak formulation corresponding to (3.1.1) is given by : Seek v ∈ L2(0, T ; H1f (Ωf )), p ∈

L2(0, T ;L2(Ωf )), g ∈ Y, u ∈ L2(0, T ; H1s(Ωs)) ∩H1(0, T ; L2(Ωs)) and h ∈ Z satisfying

ρf (vt,w)Ωf+ b(w, p) + af (v,w) + (w, pn− µf∇v · n)Γ0

= ρf (f ,w)Ωf∀ w ∈ H1

f (Ωf ) (3.2.2)

b(v, q) = 0 ∀ q ∈ L2(Ωf ) (3.2.3)

(v, s)Γ0 − (g, s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0) (3.2.4)

ρs(utt, θ)Ωs + as(u, θ) = ρs(b, θ)Ωs + (h, θ)Γ0

∀ θ ∈ H1s(Ωs) (3.2.5)

v|t=0 = v0 in Ωf (3.2.6)

u|t=0 = u0 in Ωs (3.2.7)

ut|t=0 = u1 in Ωs (3.2.8)

Next, we give a precise definition of an optimal solution. Let the admissibility set is

defined by

Uad = (v, p,g,u,h) ∈ L2(0, T ; H1f (Ωf ))× L2(0, T ;L2(Ωf ))×

Y × L2(0, T ; H1s(Ωs)) ∩H1(0, T ; L2(Ωs))× Z :

J δ(v, p,g,u,h) <∞ and (3.2.2)-(3.2.8) is satisfied (3.2.9)

23

Then (v, p, g, u, h) is called an optimal solution if there exists α ≥ 0 such that

J δ(v, p, g, u, h) ≤ J δ(v, p,g,u,h)

for all (v, p,g,u,h) ∈ Uad satisfying

‖g − g‖Y + ‖h− h‖Z ≤ α

To show the existence of optimal solutions, we introduce an auxiliary problem again.

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

v = g on Γ0

ρsutt − εµs∆ut − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

εµs∇ut · n + µs∇u · n + (λs + µs)(∇ · u)n = h on Γ0

v|t=0 = v0 in Ωf

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

(3.2.10)

Problem 2

Find (v,p,g,u,h) such that the functional Jδ is minimized subject to (3.2.10).

A weak formulation corresponding to (3.2.10) is given by: Seek v ∈ L2(0, T ; H1f (Ωf )), p ∈

L2(0, T ;L2(Ωf )), g ∈ Y, u ∈ H1(0, T ; H1s(Ωs)) and h ∈ Z satisfying

ρf (vt,w)Ωf+ b(w, p) + af (v,w) + (w, pn− µf∇v · n)Γ0

= ρf (f ,w)Ωf∀ w ∈ H1

f (Ωf ) (3.2.11)

24

b(v, q) = 0 ∀ q ∈ L2(Ωf ) (3.2.12)

(v, s)Γ0 − (g, s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0) (3.2.13)

ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ)

= ρs(b, θ)Ωs + (h, θ)Γ0 ∀ θ ∈ H1s(Ωs) (3.2.14)

v|t=0 = v0 in Ωf (3.2.15)

u|t=0 = u0 in Ωs (3.2.16)

ut|t=0 = u1 in Ωs (3.2.17)

And admissibility set be defined by

U εad = (v, p,g,u,h) ∈ L2(0, T ; H1

f (Ωf ))× L2(0, T ;L2(Ωf ))×

Y ×H1(0, T ; H1s(Ωs))× Z : J δ(v, p,g,u,h) <∞

and (3.2.11)-(3.2.17) is satisfied (3.2.18)

Using the properties of the bilinear forms we can obtain an a priori bounds for

solutions of the weak formulation (3.2.11)-(3.2.17). Let (v, p,g,u,h) satisfy (3.2.11)-

(3.2.17). From (3.2.11) and (3.2.14), we obtain

ρf (vt,w)Ωf+ b(w, p) + af (v,w)

+ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ)

= ρf (f ,w)Ωf+ ρs(b, θ)Ωs − (w, pn− µf∇v · n)Γ0 + (h, θ)Γ0

∀ w ∈ H1f (Ωf ), ∀ θ ∈ H1

s(Ωs)

or

ρf (vt,w)Ωf+ b(w, p) + af (v,w)

+ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ)

= ρf (f ,w)Ωf+ ρs(b, θ)Ωs − (w, pn− µf∇v · n)Γ0

+(w,h)Γ0 + (h, θ)Γ0 − (w,h)Γ0

25

∀ w ∈ H1f (Ωf ), ∀ θ ∈ H1

s(Ωs) (3.2.19)

Taking w = v and θ = ut in (3.2.19). Then because of (3.2.12)-(3.2.13) we have

ρf (vt,v)Ωf+ af (v,v) + ρs(utt,ut)Ωs + εa(ut,ut) + as(u,ut)

= ρf (f ,v)Ωf+ ρs(b,ut)Ωs − (g, pn− µf∇v · n)Γ0 + (g,h)Γ0

+(h,ut)Γ0 − (g,h)Γ0

This may be reduced to

ρf

2

d

dt‖v‖2

0,Ωf+Ka‖v‖2

1,Ωf+ρs

2

d

dt‖ut‖2

0,Ωs+ εKa‖ut‖2

1,Ωs+Ka

2

d

dt‖u‖2

1,Ωs

= −(g, pn− µf∇v · n− h)Γ0 + (h,ut − g)Γ0 + ρf (f ,v)Ωf+ ρs(b,ut)Ωs

≤ ‖g‖0,Γ0‖pn− µf∇v · n− h‖0,Γ0 + ‖h‖0,Γ0‖ut − g‖0,Γ0

+ρf‖f‖−1,Ωf‖v‖1,Ωf

+ ρs‖b‖0,Ωs‖ut‖0,Ωs

Integrating from 0 to t yields

ρf supt‖v‖2

0,Ωf+Ka

∫ T

0‖v‖2

1,Ωfdt+ ρs sup

t‖ut‖2

0,Ωs

+εKa

∫ T

0‖ut‖2

1,Ωsdt+Ka sup

t‖u‖2

1,Ωs

≤∫ T

0‖g‖2

0,Γ0dt+

∫ T

0‖pn− µf∇v · n− h‖2

0,Γ0dt+

∫ T

0‖h‖2

0,Γ0dt

+∫ T

0‖ut − g‖2

0,Γ0dt+ ρfC

∫ T

0‖f‖2

−1,Ωfdt+ ρsC

∫ T

0‖b‖2

0,Ωsdt

+ρf‖v0‖20,Ωf

+ ρs‖u1‖20,Ωs

+Ka‖u0‖21,Ωs

(3.2.20)

We take test functions w with ∇ ·w = 0 in equation (3.2.11), we get

ρf (vt,w)Ωf= −af (v,w)− (w, pn− µf∇v · n)Γ0 + ρf (f ,w)Ωf

∀w ∈ V

and this gives

|ρf (vt,w)Ωf| ≤ ka‖v‖1,Ωf

‖w‖1,Ωf

+ ‖w‖1,Ωf‖pn− µf∇v · n‖0,Γ0 + ρf‖f‖−1,Ωf

‖w‖1,Ωf

26

Therefore,

ρf

∫ T

0‖vt‖2

V ∗dt ≤ ka

∫ T

0‖v‖2

1,Ωfdt+

∫ T

0‖pn− µf∇v · n− h‖2

0,Γ0dt

+∫ T

0‖h‖2

0,Γ0dt+ ρf

∫ T

0‖f‖2

−1,Ωfdt (3.2.21)

From (3.2.11),

b(w, p) = −ρf (vt,w)Ωf− af (v,w)

− (w, pn− µf∇v · n)Γ0 + ρf (f ,w)Ωf∀w ∈ H1

f (Ωf )

Inf-sup condition may be used to get,

Kb

∫ T

0‖p‖2

0,Ωfdt ≤ ρf

∫ T

0‖vt‖2

V ∗dt+ ka

∫ T

0‖v‖2

1,Ωfdt

+∫ T

0‖pn− µf∇v · n‖2

0,Γ0dt+ ρf

∫ T

0‖f‖2

−1,Ωfdt (3.2.22)

Theorem 3.2.1 There exists an optimal solution (v, p, g, u, h) ∈ U εad for Problem 2.

proof: It is clear that U εad is not empty. Let (vn, pn,gn,un,hn) be a minimizing sequence

in Uad. i.e.

limn→∞

J δ(vn, pn,gn,un,hn) = inf

(v,p,g,u,h)∈Uad

J δ(v, p,g,u,h)

Thus from (3.2.18), we have that ‖gn‖ and ‖hn‖ are uniformly bounded in Y × Z.

Then since by (3.2.20) and (3.2.22) (vn, pn,un) ∈ L2(0, T ; H1f (Ωf ))×L2(0, T ;L2(Ωf ))×

H1(0, T ; H1s(Ωs)) is uniformly bounded. Thus, there exist subsequences, denoted by

(vn, pn,gn,un,hn) for simplicity, such that

vn v in L2(0, T ; H1f (Ωf ))

pn p in L2(0, T ; L2(Ωf ))

gn g in Y

un u in H1(0, T ; H1s(Ωs))

hn h in Z

27

for some (v, p, g, u, h) ∈ U εad.

Now, by passing to the limit, we have that (v, p, g, u, h) ∈ U εad satisfies (3.2.11)-

(3.2.17). Then by the fact that functional Jδ(·, ·, ·, ·, ·) is lower semi-continuous, we

conclude that

J δ(v, p, g, u, h) = inf(v,p,g,u,h)∈U ε

ad

J δ(v, p,g,u,h)

i.e. (v, p, g, u, h) is an optimal solution.

Now we take the limit as ε → 0 of optimal solutions of Problem 2 to get optimal

solutions of Problem 1.

Theorem 3.2.2 There exists an optimal solution (v, p, g, u, h) ∈ Uad for Problem 1.

The proof is the same as in the Theorem 2.2.

3.3 Convergence with vanishing penalty parameter

In the next theorem we show that optimal solutions of Problem 2 converges to the

weak solution of (2.4.16) as δ → 0.

Theorem 3.3.1 For each δ, let (vδ, pδ,gδ,uδ,hδ) denote an optimal solution of Problem

2 and (v, p, u) denote a solution of (2.4.22)-(2.4.26). Then ‖v − vδ‖1,Ωf→ 0, ‖p −

pδ‖0,Ωf→ 0, ‖u− uδ‖1,Ωs → 0, ‖ut − uδ

t‖1,Ωs → 0 as δ → 0.

proof: Let g = v|Γ0 and h = εµs∇ut · n + µs∇u · n + (λs + µs)(∇ · u)n on Γ0. For any

solution (v, p, u) of (2.4.16), we have

J δ(vδ, pδ,gδ,uδ,hδ) ≤ J δ(v, p, g, u, h)

i.e.

1

2

∫ T

0

∫Γ0

(uδt − g)2 dΓ dt+

1

2

∫ T

0

∫Γ0

(pδn− µf∇vδ · n− hδ)2dΓ dt

+δ

2

∫ T

0

∫Γ0

‖gδ‖2Y dΓ dt+

δ

2

∫ T

0

∫Γ0

‖hδ‖2ZdΓ dt

≤ J (v, p, g, u, h) +δ

2

∫ T

0

∫Γ0

‖g‖2Y dΓ dt+

δ

2

∫ T

0

∫Γ0

‖h‖2ZdΓ dt ∀δ

28

Then ‖gδ‖Y and ‖hδ‖Z are uniformly bounded, ‖uδt − g‖0,Γ0 → 0 and ‖pδn −

µf∇vδ · n − hδ‖0,Γ0 → 0 as δ → 0. We then obtain ‖vδ‖1,Ωf, ‖pδ‖0,Ωf

, ‖uδ‖1,Ωs and

‖uδt‖1,Ωs are uniformly bounded by (3.2.20) and (3.2.22). Hence, as δ → 0, there

exists a subsequence of (vδ, pδ,gδ,uδ,hδ) that converges to some (v, p, g, u, h) ∈

L2(0, T ; H1f (Ωf )) × L2(0, T ;L2(Ωf )) × Y ×H1(0, T ; H1

s(Ωs)) × Z. The fact that ‖uδt −

g‖0,Γ0 → 0 yields v = ut on Γ0, also ‖pδn − µf∇vδ · n − hδ‖0,Γ0 → 0 as δ → 0 implies

that εµs∇ut · n + µs∇u · n + (λs + µs)(∇ · u)n = pn− µf∇v · n on Γ0.

By passing to the limit, we have that (v, p, u) satisfy (2.4.22)-(2.4.26). By the unique-

ness of solutions of (2.4.22)-(2.4.26), (v, p, u) = (v, p, u) and hence theorem is proved.

The following theorem is achieved due to Theorem 2.4.2, Theorem 3.2.2 and Theorem

3.3.1.

Theorem 3.3.2 For each δ, let (vδ, pδ,gδ,uδ,hδ) denote an optimal solution of Problem

1 and (v, p, u) denote a solution of (2.3.7)-(2.3.11). Then ‖v − vδ‖1,Ωf→ 0, ‖p −

pδ‖0,Ωf→ 0, ‖u− uδ‖1,Ωs → 0, ‖ut − uδ

t‖1,Ωs → 0 as δ → 0.

3.4 Optimality system

We use the Lagrange multiplier rule to derive the first order necessary conditions

that optimal solutions must satisfy.

Let B1 = L2(0, T ;H1f (Ωf ))×L2(0, T ;L2(Ωf ))× Y ×H1(0, T ;H1

s (Ωs))×Z and B2 =

L2(0, T ;H1f (Ωf ))∗ × L2(0, T ;H1

f (Ωf ))× Y ×H1(0, T ;H1s (Ωs))

∗.

where, (·) denotes the dual space. Suppose the linear operator M : B1 → B2 denotes

the constraint operator, i.e., M : B1 → B2 is defined by M(v, p,g,u,h) = (f , z,d,b) if

and only if

∫ T

0[ρf (vt,w)Ωf

+ b(w, p) + af (v,w) + (w, pn− µf∇v · n)Γ0 ] dt

=∫ T

0ρf (f ,w)Ωf

dt ∀ w ∈ H1f (Ωf )

29

∫ T

0b(v, q) dt =

∫ T

0b(z, q) dt ∀ q ∈ L2(Ωf )∫ T

0[(v, s)Γ0 − (g, s)Γ0 ] dt =

∫ T

0(d, s)Γ0 dt ∀ s ∈ H−1/2(Γ0)

and∫ T

0[ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ)− (h, θ)Γ0 ] dt

=∫ T

0ρs(b, θ)Ωs dt ∀ θ ∈ H1

s (Ωs)

Note that the constraint equations (3.2.11)-(3.2.14) can simply be expressed as

M(v, p,g,u,h) = (f , 0, 0,b)

The operator M ′(v, p,g,u,h) ∈ L(B1, B2) may be defined as follows:

M ′(v, p,g,u,h) · (Λ, y,k, χ, l) = (f , z, d, b) if and only if∫ T

0[ρf (Λt,w)Ωf

+ b(w, y) + af (Λ,w) + (w, yn− µf∇Λ · n)Γ0 ] dt

=∫ T

0ρf (f ,w)Ωf

dt ∀ w ∈ H1f (Ωf )∫ T

0b(Λ, q) dt =

∫ T

0b(z, q) dt ∀ q ∈ L2(Ωf )∫ T

0[(Λ, s)Γ0 − (k, s)Γ0 ] dt =

∫ T

0(d, s)Γ0 dt ∀ s ∈ H−1/2(Γ0)

and∫ T

0[ρs(χtt, θ)Ωs + εa(χt, θ) + as(χ, θ)− (l, θ)Γ0 ] dt

=∫ T

0ρs(b, θ)Ωs dt ∀ θ ∈ H1

s (Ωs)

We also have that the operator Jδ ∈ L(B1, B2) may be defined by

Jδ′(v, p,g,u,h) · (Λ,y,k, χ, l) = a for (Λ,y,k, χ, l) ∈ B1

if and only if∫ T

0(ut − g, χt − k)Γ0 dt

+∫ T

0(pn− µf∇v · n− h, yn− µf∇Λ · n− l)Γ0 dt

+δ∫ T

0(g,k)Γ0 dt+ δ

∫ T

0(∇Γg,∇Γk)Γ0 dt+ δ

∫ T

0(gt,kt)Γ0 dt

+δ∫ T

0(h, l)Γ0 dt = a

30

Suppose (v, p, g, u, h) ∈ B1 is an optimal solution of Problem1. Then there exists a

nonzero Lagrange multiplier (µ, φ, τ, η) ∈ B∗2 satisfying the Euler equation.

−Jδ′(v, p,g,u,h) · (Λ,y,k, χ, l)

+ < (µ, φ, τ, η),M ′(v, p,g, t,u,h) · (Λ,y,k, χ, l) >= 0

∀(Λ,y,k, χ, l) ∈ B1 (3.4.23)

where, < ·, · > denotes the duality pairing between B2 and the dual space B∗2 .

∫ T

0[ρf (Λt, µ)Ωf

+ b(µ, y) + af (Λ, µ) + (µ, yn− µf∇Λ · n)Γ0 + b(Λ, φ)

+(Λ, τ)Γ0 − (k, τ)Γ0 + ρs(χtt, η)Ωs + εa(χt, η) + as(χ, η)− (l, η)Γ0 ] dt

=∫ T

0[(ut − g, χt − k)Γ0

+(pn− µf∇v · n− h, yn− µf∇Λ · n− l)Γ0 ] dt

+δ∫ T

0(g,k)Γ0 dt+ δ

∫ T

0(∇Γg,∇Γk)Γ0 dt+ δ

∫ T

0(gt,kt)Γ0 dt

+δ∫ T

0(h, l)Γ0 dt ∀(Λ,y,k, χ, l) ∈ B1 (3.4.24)

We may rewrite (3.4.24) in the form

−ρf (µt,Λ)Ωf+ af (Λ, µ) + b(Λ, φ) (3.4.25)

+(Λ, τ)Γ0 − (µf∇Λ · n, µ)

= −(pn− µf∇v · n− h, µf∇Λ · n) ∀ Λ ∈ H1f (Ωf )

b(µ, y) = 0 ∀ y ∈ L2(Ωf ) (3.4.26)

(µ, yn)Γ0 = (pn− µf∇v · n− h, yn)Γ0 ∀ y ∈ L20(Ωf ) (3.4.27)

µ|t=T = 0 (3.4.28)

ρs(ηtt, χ)Ωs + εa(χt, η) + as(χ, η)

= −(utt − gt, χ)Γ0 ∀ χ ∈ H1s (Ωs) (3.4.29)

η|t=T = 0 (3.4.30)

ηt|t=T = 0 (3.4.31)

31

(k, τ)Γ0 = (ut − g,k)Γ0 − δ(g,k)Γ0 (3.4.32)

−δ(∆Γg,∆Γk)Γ0 + δ(gtt,k)Γ0 ∀ k ∈ Y

(l, η)Γ0 = (pn− µf∇v · n− h, l)Γ0 − δ(h, l)Γ0 ∀ l ∈ Z (3.4.33)

Thus, solutions of the optimal problem are determined by solving the system (3.2.11)-

(3.2.17) and (3.4.25)-(3.4.33). This system of equations is called the optimality system.

We may replace (3.4.25) and (3.4.32) by

−ρf (µt,Λ)Ωf+ af (Λ, µ) + b(Λ, φ) (3.4.34)

+(Λ, φn− µf∇µ · n)Γ0 = 0 ∀ Λ ∈ H1f (Ωf )

(k, φn− µf∇µ · n)Γ0 = (ut − g,k)Γ0 − δ(g,k)Γ0 (3.4.35)

−δ(∆Γg,∆Γk)Γ0 + δ(gtt,k)Γ0 ∀ k ∈ Y

32

The optimality system is a weak formulation corresponding to

ρfvt +∇p− µf∆v = ρf f in Ωf

∇ · v = 0 in Ωf

v = 0 on Γf

v = g on Γ0

ρsutt − εµs∆ut − µs∆u− (λs + µs)∇(∇ · u) = ρsb in Ωs

u = 0 on Γs

εµs∇ut · n + µs∇u · n + (λs + µs)(∇ · u)n = h on Γ0

v|t=0 = v0 in Ωf

u|t=0 = u0 in Ωs

ut|t=0 = u1 in Ωs

−ρfµt +∇φ− µf∆µ = 0 in Ωf

∇ · µ = 0 in Ωf

µ = 0 on Γf

µ = pn− µf∇v · n− h on Γ0

ρsηtt − εµs∆ηt − µs∆η − (λs + µs)∇(∇ · η) = 0 in Ωs

η = 0 on Γs

εµs∇ηt · n + µs∇η · n + (λs + µs)(∇ · η)n = −(utt − gt) on Γ0

µ|t=T = 0 in Ωf

η|t=T = 0 in Ωs

ηt|t=T = 0 in Ωs

(1 + δ)g − δ∆Γg − δgtt = −φn + µf∇µ · n + ut on Γ0

(1 + δ)h = (pn− µf∇v · n− η) on Γ0

(3.4.36)

33

3.5 Sensitivity derivatives

The optimality system (3.2.11)-(3.2.17) and (3.4.25)-(3.4.33) may also be derived

using sensitivity derivatives instead of the Lagrange multiplier rule. The first derivatives

∂Jδ

∂g,∂Jδ

∂hof Jδ are defined through their actions on variations g and h as follows:

<∂Jδ

∂g, g >=

∫ T

0(ut − g,−g)Γ0dt (3.5.37)

+∫ T

0(pn− µf∇v · n− h, pn− µf∇v · n)Γ0dt

+δ∫ T

0(g, g)Γ0dt+ δ

∫ T

0(∇Γg,∇Γg)Γ0dt

+δ∫ T

0(gt, gt)Γ0dt (3.5.38)

where p, v are solutions of

ρf (vt,w)Ωf+ b(w, p) + af (v,w) (3.5.39)

+(w, pn− µf∇v · n)Γ0 = 0 ∀ w ∈ H1f (Ωf )

b(v, q) = 0 ∀ q ∈ H(Ωf ) (3.5.40)

(v, s)Γ0 − (g, s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0) (3.5.41)

v|t=0 = 0 (3.5.42)

Setting Λ = v , yn = pn− µf∇v · n in (3.4.34),(3.4.27) and w = µ in (3.5.39) and

from (3.5.37)

<∂Jδ

∂g, g >=

∫ T

0(ut − g,−g)Γ0dt

+∫ T

0(g, φn− µf∇µ · n)Γ0dt+ δ

∫ T

0(g, g)Γ0dt (3.5.43)

−δ∫ T

0(∆Γg, g)Γ0dt− δ

∫ T

0(gtt, g)Γ0dt

<∂Jδ

∂h, h >=

∫ T

0(ut − g, ut)Γ0dt

+∫ T

0(pn− µf∇v · n− h,−h)Γ0dt+ δ

∫ T

0(h, h)Γ0dt (3.5.44)

34

where u is a solution of

ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ) = (h, θ)Γ0 ∀ θ ∈ H1s (Ωs) (3.5.45)

u|t=0 = 0 (3.5.46)

ut|t=0 = 0 (3.5.47)

Setting χ = u in (3.4.29) and θ = η in (3.5.45) and from (3.5.44)

<∂Jδ

∂h, h >=

∫ T

0(h, η)Γ0dt

+∫ T

0(pn− µf∇v · n− h,−h)Γ0dt+ δ

∫ T

0(h, h)Γ0dt (3.5.48)

Thus the first order necessary conditions ∂Jδ∂g

= 0 and ∂Jδ∂h

= 0 yield that

(1 + δ)g − δ∆Γg − δgtt = −φn + µf∇µ · n + ut on Γ0

(1 + δ)h = pn− µf∇v · n− η on Γ0

which are the same as in (3.4.36).

Note that equations (3.5.43) and (3.5.48) give an explicit formula for the gradient of

Jδ, i.e.,

∂Jδ

∂g=

∫ T

0[(1 + δ)g − δ∆Γg − δgtt + φn− µf∇µ · n− ut]dt (3.5.49)

∂Jδ

∂h=

∫ T

0[(1 + δ)h− pn + µf∇v · n + η]dt (3.5.50)

3.6 Gradient method

In this section, we study a gradient method to solve the optimization system (3.2.11)-

(3.2.17) and (3.4.25)-(3.4.33). The simple gradient method we consider is defined as

follows.

35

Given a starting guess g(0) and h(0)

g(n+1)

h(n+1)

=

g(n)

h(n)

− α

∂Jδ∂g

∂Jδ∂h

for n = 1, 2, · · · ,

where α is a step size. Combining with (3.5.49) and (3.5.50) yields, for n = 1, 2, · · · g(n+1)

h(n+1)

=

g(n)

h(n)

− α

∫ T0 [(1 + δ)g − δ∆Γg − δgtt + φn− µf∇µ · n− ut]dt∫ T0 [(1 + δ)h− pn + µf∇v · n + η]dt

Algorithm 3.6.1 1. Choose g(0) and h(0).

2. For n=0,1,2,· · ·,

(a) determine v(n), p(n) and u(n) from

ρf (v(n)t ,w)Ωf

+ b(w, p(n)) + af (v(n),w)

+(w, p(n)n− µf∇v(n) · n)Γ0 = ρf (f ,w)Ωf∀ w ∈ H1

f (Ωf )

b(v(n), q) = 0 ∀ q ∈ H2(Ωf )

(v(n), s)Γ0 − (g(n), s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0)

v(n)|t=0 = v0 in Ωf

ρs(u(n)tt , θ)Ωs + εa(u

(n)t , θ) + as(u

(n), θ) = ρs(b, θ)Ωs + (h(n), θ)Γ0

∀ θ ∈ H1s (Ωs)

u(n)|t=0 = u0 in Ωs

u(n)t |t=0 = u1 in Ωs

36

(b) determine µ(n),φ(n) and η(n) from

−ρf (µ(n)t ,Λ)Ωf

+ af (Λ, µ(n)) + b(Λ, φ(n))

+(Λ, φnn− µf∇µ(n) · n)Γ0 = 0 ∀ Λ ∈ H1f (Ωf )

b(µ(n), q) = 0 ∀ q ∈ L2(Ωf )

(µ(n), s)Γ0 = (p(n)n− µf∇v(n) · n− h(n), s)Γ0 ∀ s ∈ H−1/2(Γ0)

µ(n)|t=T = 0

ρs(η(n)tt , χ)Ωs + εa(χt, η

(n)) + as(χ, η(n)) = −(u

(n)tt − g

(n)t , χ(n))Γ0

∀ χ ∈ H1s (Ωs)

η(n)|t=T = 0

η(n)t |t=T = 0

(c) determine g(n+1) and h(n+1) from

g(n+1) = g(n)

−α∫ T0 [(1 + δ)g(n) − δ∆Γg

(n) − δg(n)tt + φ(n)n− µf∇µ(n) · n− u

(n)t ]dt

h(n+1) = h(n) − α∫ T0 [(1 + δ)h(n) − p(n)n + µf∇v(n) · n + η(n)]dt

The following result is useful to determining sufficient conditions for the convergence

of the gradient method.

Theorem 3.6.1 Let X be a Hilbert space equipped with the inner product (·, ·)X and

norm ‖ · ‖X . Suppose M is a functional on x such that

1. M has a local minimum at x and is twice differentiable in an open ball B centered

at x;

2. | <M′′(u), (x, y) > | ≤M‖x‖X‖y‖Y , ∀u ∈ B, x ∈ X, y ∈ Y ;

3. | <M′′(u), (x, x) > | ≥ m‖x‖2X , ∀u ∈ B, x ∈ X,

37

where M and m are positive constants. Let R denote the Riesz map. Choose x(0)

sufficiently close to x and choose a sequence ρn such that 0 < ρ∗ ≤ ρn ≤ ρ∗ < 2m/M2.

Then the sequence x(n) defined by

x(n) = x(n−1) − ρnRM′(x(n−1)), for n = 1, 2, · · · ,

converges to x.

We examine the second derivatives of J to determine the constants M and m.

<∂2J∂g2

, (g,g) >=∫ T

0(g, g)Γ0dt

+∫ T

0(pn− µf∇v · n, pn− µf∇v · n)Γ0dt

+δ∫ T

0(g, g)Γ0dt+ δ

∫ T

0(∇Γg,∇Γg)Γ0dt+ δ

∫ T

0(gt, gt)Γ0dt

<∂2J∂g∂h

, (g,h) >=∫ T

0(−g,ut)Γ0dt

+∫ T

0(pn− µf∇v · n,−h)Γ0dt

<∂2J∂h∂g

, (h,g) >=∫ T

0(ut,−g)Γ0dt

+∫ T

0(−h, pn− µf∇v · n)Γ0dt

<∂2J∂h2

, (h,h) >=∫ T

0(ut, ut)Γ0dt+ (1 + δ)

∫ T

0(h, h)Γ0dt

where, v, p,u, v, p and u are solutions of

ρf (vt,w)Ωf+ b(w, p) + af (v,w)

+(w, pn− µf∇v · n)Γ0 = 0 ∀ w ∈ H1f (Ωf )

b(v, q) = 0 ∀ q ∈ L2(Ωf )

(v, s)Γ0 − (g, s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0)

v|t=0 = 0

(3.6.51)

38

ρf (vt,w)Ωf+ b(w, p) + af (v,w)

+(w, pn− µf∇v · n)Γ0 = 0 ∀ w ∈ H1f (Ωf )

b(v, q) = 0 ∀ q ∈ L2(Ωf )

(v, s)Γ0 − (g, s)Γ0 = 0 ∀ s ∈ H−1/2(Γ0)

v|t=0 = 0

(3.6.52)

ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ) = (h, θ)Γ0 ∀ θ ∈ H1s (Ωs)

u|t=0 = 0

ut|t=0 = 0

(3.6.53)

ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ) = (h, θ)Γ0 ∀ θ ∈ H1s (Ωs)

u|t=0 = 0

ut|t=0 = 0

(3.6.54)

Then,

(g h)

∂2J∂g2

∂2J∂g∂h

∂2J∂h∂g

∂2J∂h2

g

h

= <

∂2J∂g2

, (g,g) > + <∂2J∂g∂h

, (g,h) > + <∂2J∂h∂g

, (h,g) > + <∂2J∂h2

, (h,h) >

=∫ T

0(g, g)Γ0dt+

∫ T

0(pn− µf∇v · n, pn− µf∇v · n)Γ0dt

+δ∫ T

0(g, g)Γ0dt+ δ

∫ T

0(∇Γg,∇Γg)Γ0dt+ δ

∫ T

0(gt, gt)Γ0dt

+∫ T

0(−g,ut)Γ0dt+

∫ T

0(pn− µf∇v · n,−h)Γ0dt

+∫ T

0(ut,−g)Γ0dt+

∫ T

0(−h, pn− µf∇v · n)Γ0dt

+∫ T

0(ut, ut)Γ0dt+ (1 + δ)

∫ T

0(h, h)Γ0dt

=∫ T

0(ut − g, ut − g)Γ0dt

39

+∫ T

0(pn− µf∇v · n− h, pn− µf∇v · n− h)Γ0dt

+δ∫ T

0(g, g)Γ0dt+ δ

∫ T

0(∇Γg,∇Γg)Γ0dt+ δ

∫ T

0(gt, gt)Γ0dt

+δ∫ T

0(h, h)Γ0dt

We show that ∣∣∣∣∣∣∣∣(g, h)

∂2J∂g2

∂2J∂g∂h

∂2J∂h∂g

∂2J∂h2

g

h

∣∣∣∣∣∣∣∣ ≤M‖x‖‖y‖

where x = (g, h)T , y = (g,h)T , ‖x‖ =√‖g‖2

Y + ‖h‖2Z and ‖y‖ =

√‖g‖2

Y + ‖h‖2Z

First, we obtain bounds for v and p. Let v, p satisfy (3.6.51), which is a weak

formulation of

ρfvt +∇p− µf∆v = 0 in Ωf

∇ · v = 0 in Ωf

v = g on Γ

v|t=0 = 0 in Ωf

(3.6.55)

where, Γ = Γf ∪ Γ0 and

g =

g on Γ0

0 on Γf

We introduce a space

Hs∆(Ωf ) = v ∈ Hs(Ωf ) : ∆v ∈ L2(Ωf )

with a norm

‖v‖s∆,Ωf= ‖v‖s,Ωf

+ ‖∆v‖0,Ωf

Then, for g ∈ L2(0, T ; H1(Γ)) ∩H1(0, T ; L2(Γ)), there is an extension R0g ∈

L2(0, T ; H3/2∆ (Ωf ))∩H1(0, T ; H

1/2∆ (Ωf )) such that∇·(R0g) = 0 in Ωf and R0g = g on Γ.

40

And

‖R0g‖3/2∆,Ωf+ ‖R0gt‖1/2∆,Ωf

≤ ‖g‖1,Γ + ‖gt‖0,Γ. (3.6.56)

Write v = v +R0g, then we have

ρf vt +∇p− µf∆v = −ρfR0gt + µf∆R0g in Ωf

∇ · v = 0 in Ωf

v = 0 on Γ

v|t=0 = −ρfR0g|t=0 in Ωf

(3.6.57)

Notice that −ρfR0gt + µf∆R0g ∈ L2(Ωf ) and we show that R0g|t=0 ∈ H1(Ωf ).

[X, Y ]θ denotes the interpolation space equipped with a norm

‖u‖[X,Y ]θ = ‖Λ1−θu‖Y

where Λ : Y → Y is an operator such that ‖u‖X = ‖Λu‖Y . And define a space W 1(0, T )

as

W 1(0, T ) = u ∈ L2(0, T ;X), ut ∈ L2(0, T ;Y )

Theorem 3.6.2 (Trace theorem) There exists a bounded operator γ0 : W 1(0, T ) →

[X, Y ]1/2 such that γ0u = u|t=0 and ‖γ0u‖[X,Y ]1/2≤ C‖u‖W 1(0,T )

proof: See Vivsik, Fursikov [40].

Lemma 3.6.3 If X = H3/2(Ωf ) and Y = H1/2(Ωf ) then [X, Y ]1/2 = H1(Ωf )

proof:

(Su1,u2)1/2,Ωf=

∫Ωf

(1 + |ξ|2)1/2Su1u2dξ

(u1,u2)3/2,Ωf=

∫Ωf

(1 + |ξ|2)3/2u1u2dξ

41

(Su1,u2)1/2,Ωf= (u1,u2)3/2,Ωf

implies S = 1 + |ξ|2 and hence Λ = (1 + |ξ|2)1/2 i.e.

‖Λu‖1/2,Ωf= ‖u‖3/2,Ωf

. By the definition of the interpolation space,

‖u‖2[X,Y ]1/2

= ‖Λ1/2u‖21/2,Ωf

= ‖u‖21,Ωf

Therefore, [X,Y ]1/2 = H1(Ωf )

If X = H3/2(Ωf ) and Y = H1/2(Ωf ) then we may rewrite the Trace Theorem as

follows :

There exists a bounded operator γ0 : W 1(0, T ) → H1(Ωf ) such that γ0u = u|t=0 and

‖γ0u‖21,Ωf

≤ C(∫ T0 ‖u‖2

3/2,Ωfdt+

∫ T0 ‖ut‖2

1/2,Ωfdt)

Since R0g ∈ L2(0, T ; H3/2∆ (Ωf )) ∩ H1(0, T ; H

1/2∆ (Ωf )), we have R0g|t=0 ∈ H1(Ωf ))

and

‖R0g|t=0‖21,Ω ≤ C(

∫ T

0‖R0g‖2

3/2,Ωfdt+

∫ T

0‖R0gt‖2

1/2,Ωfdt) (3.6.58)

Then (3.6.57) is an evolution Stokes equation and hence we obtain

v ∈ L2(0, T ; H2(Ωf ))

vt ∈ L2(0, T ; L2(Ωf ))

p ∈ L2(0, T ;H1(Ωf ))

Moreover,

∫ T

0‖vt‖2

0,Ωfdt ≤ ρ2

f

∫ T

0‖R0gt‖2

0,Ωfdt

µ2f

∫ T

0‖∆R0g‖2

0,Ωfdt+ ‖R0g|t=0‖2

1,Ω

≤ C(‖g‖21,Γ + ‖gt‖2

0,Γ)

by (3.6.56) and (3.6.58).

We then come back to (3.6.57) and we apply the regularity theorem of stationary

42

case. For almost every t in [0, T ],

∇p− µf∆v = −ρf vt − ρfR0gt + µf∆R0g in Ωf

∇ · v = 0 in Ωf

v = 0 on Γ

so that v(t) ∈ H2(Ω) and p(t) ∈ H1(Ω). Moreover,

∫ T

0‖v‖2

2,Ωfdt+

∫ T

0‖p‖2

1,Ωfdt

≤ C(ρ2f

∫ T

0‖vt‖2

0,Ωfdt+ ρ2

f

∫ T

0‖R0gt‖2

0,Ωfdt+ µ2

f

∫ T

0‖∆R0g‖2

0,Ωfdt)

≤ C(‖g‖21,Γ + ‖gt‖2

0,Γ)

Now we get bound for v.

∫ T

0‖v‖2

3/2∆,Ωfdt ≤

∫ T

0‖v‖2

3/2∆,Ωfdt+

∫ T

0‖R0g‖2

3/2∆,Ωfdt

≤ 2(‖v‖23/2,Ωf

dt+ ‖∆v‖20,Ωf

dt) +∫ T

0‖R0g‖2

3/2∆,Ωfdt

≤ C(‖g‖21,Γ + ‖gt‖2

0,Γ)

Then

∫ T

0‖pn‖2

0,Γ0dt ≤

∫ T

0‖p‖2

1,Ωfdt

≤ C(‖g‖21,Γ + ‖gt‖2

0,Γ) = C‖g‖2Y

and

∫ T

0‖∇v · n‖2

0,Γ0dt ≤

∫ T

0‖v‖2

3/2∆,Ωfdt

≤ C(‖g‖21,Γ + ‖gt‖2

0,Γ) = C‖g‖2Y

Now, we obtain a bound for u. Suppose u is a solution of (3.6.53). Then

ρs(utt, θ)Ωs + εa(ut, θ) + as(u, θ) = (h, θ)Γ0 ∀ θ ∈ H1s (Ωs) (3.6.59)

43

Setting θ = ut in (3.6.59) to get

suptρs‖ut‖2

0,Ωs+ εKa

∫ T

0‖ut‖2

1,Ωsdt+ sup

tKa‖u‖2

1,Ωs

≤ C∫ T

0‖h‖2

0,Γ0dt = C‖h‖2

Z

We determine the constants M using bounds we have obtained.∣∣∣∣∣∣∣∣(g, h)

∂2J∂g2

∂2J∂g∂h

∂2J∂h∂g

∂2J∂h2

g

h

∣∣∣∣∣∣∣∣

= |∫ T

0(ut − g, ut − g)Γ0dt

+∫ T

0(pn− µf∇v · n− h, pn− µf∇v · n− h)Γ0dt

+δ∫ T

0(g, g)Γ0dt+ δ

∫ T

0(∇Γg,∇Γg)Γ0dt+ δ

∫ T

0(gt, gt)Γ0dt

+δ∫ T

0(h, h)Γ0dt|

≤ 2(∫ T

0‖ut‖2

1,Ωsdt+

∫ T

0‖g‖2

0,Γ0dt)1/2(

∫ T

0‖ut‖2

1,Ωsdt+

∫ T

0‖g‖2

0,Γ0dt)1/2

+2(∫ T

0‖pn‖2

0,Γ0dt+ µ2

f

∫ T

0‖∇v · n‖2

0,Γ0dt+

∫ T

0‖h‖2

0,Γ0dt)1/2

(∫ T

0‖pn‖2

0,Γ0dt+ µ2

f

∫ T

0‖∇v · n‖2

0,Γ0dt+

∫ T

0‖h‖2

0,Γ0dt)1/2

+δ(∫ T

0‖g‖2

0,Γ0dt)1/2(

∫ T

0‖g‖2

0,Γ0dt)1/2

+δ(∫ T

0‖∇Γg‖2

0,Γ0dt)1/2(

∫ T

0‖∇Γg‖2

0,Γ0dt)1/2

+δ(∫ T

0‖gt‖2

0,Γ0dt)1/2(

∫ T

0‖gt‖2

0,Γ0dt)1/2

+δ(∫ T

0‖h‖2

0,Γ0dt)1/2(

∫ T

0‖h‖2

0,Γ0dt)1/2

≤ 2(C‖h‖2Z + ‖g‖2

Y )1/2(C‖h‖2Z + ‖g‖2

Y )1/2

+2(C‖g‖2Y + µ2

fC‖g‖2Y + ‖h‖2

Z)1/2(C‖g‖2Y + µ2

fC‖g‖2Y + ‖h‖2

Z)1/2

+δ‖g‖Y ‖g‖Y + δ‖h‖Y ‖h‖Y

≤ M‖x‖‖y‖

where, M = 2C + 2(1 + µ2f )C + δ.

44

Setting g = g and h = h to determine the constant m.

(g,h)

∂2J∂g2

∂2J∂g∂h

∂2J∂h∂g

∂2J∂h2

g

h

=

∫ T

0

∫Γ0

(ut − g)2 dΓ dt+∫ T

0

∫Γ0

(pn− µf∇v · n− h)2dΓ dt

+δ∫ T

0

∫Γ0

g2 dΓ dt+ δ∫ T

0

∫Γ0

(∇Γg)2 dΓ dt

+δ∫ T

0

∫Γ0

g2t dΓ dt+ δ

∫ T

0

∫Γ0

h2 dΓ dt

= J (v, p,g,u,h) + δ‖g‖2Y + δ‖h‖2

Z ≥ δ‖x‖2

where m = δ.

This proves the convergence of the Gradient method.

45

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