Luận án tiến sĩ toán học Data Assimilation in Heat Conduction

Chúng tôi sử dụng phương pháp biến phân nghiên cứu bài toán ngược này bằng cách cực tiểu hóa các phiếm hàm chỉnh.. Chúng tôi chứng minh rằng các phiếm hàm này là khả vi Fréchet và đưa ra

EDUCATION AND TRAINING SCIENCE AND TECHNOLOGY

THESIS FORTHE DEGREE OF

HANOI  2017

EDUCATION AND TRAINING SCIENCE AND TECHNOLOGY

Sp y: Dierential and Integral Equations

Supervisor: PROF DR HABIL ĐINH NHO HÀO

HANOI  2017

BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC

VÀ CÔNG NGHỆ VIỆT NAM

VIỆN TOÁN HỌC

NGUYỄN THỊ NGỌC OANH

ĐỒNG HÓA SỐ LIỆU TRONG TRUYỀN NHIỆT

Chuyên ngành: Phương trình Vi phân và Tích phân

Mã số: 62 46 01 03

LUẬN ÁN TIẾN SĨ TOÁN HỌC

Người hướng dẫn khoa học:

GS TSKH ĐINH NHO HÀO

HÀ NỘI – 2017

I first learned about inverse and ill-posed problems when I met Professor Đinh Nho Hào in

2007, my final year of bachelor’s study I have been extremely fortunate to have a chance

to study under his guidance since then I am deeply indebted to him not only for his supervision, patience, encouragement and support in my research, but also for his precious advices in life.

I would like to express my special appreciation to Professor Hà Tiến Ngoạn, Professor Nguyễn Minh Trí, Doctor Nguyễn Anh Tú, the other members of the seminar at Department

of Differential Equations and all friends in Professor Đinh Nho Hào’s group seminar for their valuable comments and suggestions to my thesis I am very grateful to Doctor Nguyễn Trung Thành (Iowa State University) for his kind help on MATLAB programming.

I would like to thank the Institute of Mathematics for providing me with such an excellent study environment.

Furthermore, I would like to thank the leaders of College of Sciences, Thai Nguyen versity, the Dean board as well as to all of my colleagues at the Faculty of Mathematics and Informatics for their encouragement and support throughout my PhD study.

Uni-Last but not least, I could not have finished this work without the constant love and unconditional support from my parents, my parents-in-law, my husband, my little children and my dearest aunt I would like to express my sincere gratitude to all of them.

are studied We reformulate these inverse problems asvariationalproblems of minimizing

appropriate mist We prove that these are F het dierentiable

and deriveaformulafortheir gradientvia adjointproblems The problems are rst

in variables by the nite methodand the variationalproblems

varia-tional problems to the solution of the tinuous ones is proved To solve the problems

n ,we further them in time by the splittingmethod It isproved that

gradient are derived via adjoint problems The problems are then solved by the

gradient method and the n algorithms are tested on As a

by-pro of the variational method based on algorithm, we suggest a simple

methodtodemonstrate the ill-posedness

Tóm tắt

Các bài toán xác định điều kiện ban đầu trong phương trình parabolic từ quan sát tại thời điểm cuối, từ quan sát tích phân bên trong, và từ quan sát biên đã được nghiên cứu Chúng tôi sử dụng phương pháp biến phân nghiên cứu bài toán ngược này bằng cách cực tiểu hóa các phiếm hàm chỉnh Chúng tôi chứng minh rằng các phiếm hàm này là khả vi Fréchet và đưa ra công thức gradient của chúng thông qua các bài toán liên hợp Trước tiên, sử dụng phương pháp sai phân hữu hạn để rời rạc hóa bài toán thuận và bài toán liên hợp tương ứng theo các biến không gian Chúng tôi chứng minh sự hội tụ của nghiệm của bài toán biến phân rời rạc tới nghiệm của bài toán biến phân liên tục Để giải số bài toán, chúng tôi tiếp tục rời rạc bài toán theo biến thời gian bằng phương pháp sai phân phân

rã (phương pháp splitting) Chúng tôi cũng chứng minh được rằng các phiếm hàm rời rạc này là khả vi Fréchet và đưa ra công thức gradient của chúng thông qua bài toán liên hợp rời rạc Sau đó chúng tôi sử dụng phương pháp gradient liên hợp để giải và các thuật toán

số được thử nghiệm trên máy tính Ngoài ra, như một sản phẩm phụ của phương pháp biến phân, dựa trên thuật toán Lanczos, chúng tôi đề xuất một phương pháp đơn giản để minh họa tính đặt không chỉnh của bài toán.

This work has been completed at Institute of Mathematics, Vietnam Academy of Science and Technology under the supervision of Prof Dr Habil Đinh Nho Hào I declare hereby that the results presented in it are new and have never been published elsewhere.

Author: Nguyen Thi Ngoc Oanh

2.1 Example1: Singularvalues 52

2.2 Example2: results: (a) v;(b)estimatedone; point-wiseerror; (d)the ofv| x 1 =1/2 andits (thedashed e:the the solid e: theestimated 53

2.3 Example3: result: (a) v;(b)estimatedone; point-wiseerror; (d)the ofv| x 1 =1/2 andits (thedashed e:the the solid e: theestimated 54

2.4 Example4: result: (a) v;(b)estimatedone; point-wiseerror; (d)the ofv| x 1 =1/2 andits (thedashed e:the the solid e: theestimated 55

2.5 Example5: result: (a) v;(b)estimatedone; point-wiseerror; (d)the ofv| x 1 =1/2 andits (thedashed e:the the solid e: theestimated 56

3.1 Example1 Singularvalues: threeobservationsandvarioustimeintervalsofobservations 68

3.2 Example2: resultsfor (a)3uniform observationpointsin(0, 0.5),errorinL 2 -norm= 0.006116;(b)3uniformobservationpointsin(0.5, 1),errorinL 2-norm= 0.006133; 3uniform observationpoints in(0.25, 0.75), theerror inL 2-norm= 0.0060894; (d)3 uniform observationpointsinΩ,theerrorinL 2-norm= 0.0057764 69

3.3 resultofExample3: (a)τ = 0.01;(b)τ = 0.05; τ = 0.1;(d)τ = 0.3 70

3.4 resultofExample4: (a)τ = 0.01;(b)τ = 0.05; τ = 0.1;(d)τ = 0.3 72

3.5 resultofExample5: (a)τ = 0.01;(b)τ = 0.05; τ = 0.1;(d)τ = 0.3 73

3.6 Example6 results: (a) initial v; (b) of v; point-wiseerror;(d)the ofv| x 1 =1/2 |andits 74

3.7 Example7 results: (a) initial v; (b) of v; point-wiseerror;(d)the ofv| x 1 =1/2 |andits 76

3.8 Example8 results: (a) initial v; (b) of v; point-wiseerror;(d)the ofv| x 1 =1/2 |andits 77

4.2 Example2,3,4:1DProblem: resultsforsmooth, tinuousand tinuous

initial 88

4.3 Example5: initial (left)andits (right) 89

4.4 Example5 tinue): Error (left) andthe v ofthe initial and its

alongtheinterval[(0.5, 0), (0.5, 1)](right) 89

4.5 Example6: initial (left)andits (right) 90

4.6 Example6 tinue): Error(left)andthe ofthe initial andits

tionalongtheinterval[(0.5, 0), (0.5, 1)](right) 90

4.7 Example7: initial (left)andits (right) 90

4.8 Example7 tinue): Error(left)andthe ofthe initial andits

tionalongtheinterval[(0.5, 0), (0.5, 1)](right) 91

3.1 Example3: Behaviorofthealgorithmwithdierentstartingpointsofobservationτ 71

3.2 Example6: Behavior of the algorithm whenthe numberof observations and the positions of

observationsvary(N = 4) 75

3.3 Example6: Behavior of the algorithm whenthe numberof observations and the positions of

observationsvary(N = 9) 75

i

1.1 problem 9

1.2 Adjointproblem 12

1.3 Finite method forone-dimensional problems 14

1.3.1 inthe variable 15

1.3.2 intime 17

1.4 Finite method formulti-dimensional problems 18

1.4.1 Interpolationsof grid 19

1.4.2 in variables and the v of the nite heme 22

1.5 Approximationof the variationalproblems 39

1.6 algorithmfor approximating singularvalues 43

Chapter 2 Data assimilation by the nal time observations 44 2.1 Problemsetting and the variationalmethod 44

2.2 of the variationalproblem in variable 47

2.3 Full of the variationalproblem 47

2.3.1 The gradient of the ob e 48

2.3.2 Conjugate gradient method 50

2.4 results 51

2.4.1 Approximationsof the singularvalues 51

2.4.2 examples for two-dimensional problems 52

Chapter 3 Data assimilation by the integral observations 57 3.1 Problemsetting and the variationalmethod 58

3.2 of the variationalproblem in variable 63

3.3 Full of the variationalproblem 64

3.4 example 67

3.4.1 One-dimensional n examples 68

3.4.2 Two-dimensional n examples 73

Chapter 4 Data assimilation by the boundary observations 78 4.1 Problemsetting and the variationalmethod 78

4.2 of the variationalmethodin variables 83

4.3 Full of the variational problem and the gradient method 84

4.4 example 86

4.4.1 example in one-dimensional 87

4.4.2 example inthe multi-dimensional 88

The author's related to the thesis 93

The of anevolution pro requiresits initial his unfortunately

not always a ailable or given in Data assimilation is the pro of

model initial from measured observations and the rst guess

eld in bination with the system Data assimilation is extensively used

in meteorology, o y, weather [2,3, 45, 66, 71℄, environmental pollution

[8, 55, 63℄, image pro [6℄, industrial pro [4, 5, 9, 24℄, For surveys of

methods in data assimilation, we refer to [2,10, 11, 15, 16, 17, 37, 46, 47, 48, 57, 64,65,

66,67, 71,73,74, 75℄, and the therein

Supposethatthepro under ismodelledbyasystemofevolutionequations

dU

where U is the v representing the state variablesthat we want to A is an

operator in the variables and F is the v of exterior onthesystem The goal of is to nd a good approximation to U during a period oftime of length T The problemwe are with is that we donot know the initialdatafor U before some time moment T 0 for the solution of the modelfrom T 0 on However, we observe (measure) U somehow, say, by CU with C being alinear operator The data assimilationproblem is to determine an approximation of the

initial atatime beforeT 0 frommeasurementsand thenuse it tosolvethe abo esystem for This problem is unfortunately ill-posed A problem is said to be

well-posedin senseof Hadamardif the following are satised[20℄: i)

There is a solution of the problem ii) Uniqueness: The solution is unique iii) Stability:

The solution tinuously depends on the data (in some appropriate topologies) If at

least one of the abo e is not fullled, the problem is said to be ill-posed (or

improperly posed) Hadamard thought that h problems have no ph meaning

However, many important problems in are ill-posed and there have been a lot of

works devoted to their study (see [2, 3, 4, 5, 7, 8, 9, 14, 19, 24, 31, 36, 45, 68, 72, 80℄

and the therein) The ill-posedness of a problem serious troubles by

making n methods unstable, that is asmall error in the data may

arbitrarilylargeerrors initssolution Tikhonov A.N.[78℄ in1943realizedthat unstability

results from ks information, and to restore stability we should impose some a priori

Tikhonovthenpointedoutthepossibilityofndingstablesolutionstoill-posed

problems Theimp ofill-posedproblemshasbeenthenrealizedbyLavrent'evM.M

the foundersof the theory of ill-posedproblems In 1963, Tikhonov [79, 80℄ published his

regularizationmethodand then inverseand ill-posedproblems b an

Thisthesis isdevotedtodataassimilationinheat namelyitaimsat

determin-ing the initial of the system (0.1) heat using three types

ofobservation: 1)observationatthenaltimemoment,2)interior(integral)observations,

3) boundary observations We nowformulateour problems more

LetΩbeanopenboundeddomaininR n , n ≥ 1,withboundary∂Ω DenoteQ = Ω×(0, T ]

with T > 0 being given, S = ∂Ω × (0, T ] Let

is thedire problem Itis proved that thereexists aweak solution(thedenition of

h will be given in Chapter 1) to these problems [24, 82℄ The inverse problem (data

assimilation) in this thesis is that of determining the initial v fromone ofthe abo ethree types ofobservation: Denotingthe solutionto(0.8)(0.10)or(0.8),

(0.9), (0.11)by u(v) toemphasize itsdep onthe initial v and supposing

that we observe u by Cu(v) The inverse problem is todetermine v when Cu(v)is given,say, by z In other words, we have to solve the equation

This problem is ill-posed as we will see that in our hosen the operator mapping

v to Cu(v) is However, its degree of ill-posedness is not an easytask Denoting the solutionto(0.8)(0.10)(or(0.8),(0.9), (0.11)) withv ≡ 0 by

o

u,weseethat the operator from v to Cv := Cu(v) − C u o is bounded and linear Thus, instead ofstudying the equation (0.12),we have todeal with the linear operator equation

The behavior of singular values of C (or eigenvalues of C ∗ C) theill-posedness of the problem [19℄ However, up to now there are very few results on this

question in inverse problems for partial dierential equations Some has

been obtained, but only in very simple [8,19,51℄ In this thesis, as aby-pro of

thevariationalmethodforndingv,weproposean hemeforestimatingsingularvalues of C h we willpresent below

To nd v, we minimizethe mist

with resp to v ∈ L 2 (Ω), instead Here, v ∗

is an a priori estimate to v, k · k H is thenormofanappropriateHilbert H andγ > 0 theregularizationparameter Itwillbeproved that there exists aunique solutionto this optimizationproblem,the J γ

isF het dierentiableanditsgradient be viaanadjointproblem Tosolve

the problemn ,we shall apply the splittingnite methodto

theoptimizationproblemsandprovethe v ofthemethod Thereasonwe hoose

the splitting nite method is that this method is easy for ding and it splits

multi-dimensional problems into a of one-dimensional problems and it is

very fast We note that one the problems by the nite element method,

however, the tsinour problems dependontime, itiseasiertouse the nite

isnot easy toanalyzethe behaviourof itssingularvalues However, aswe

∇J 0 (v) = C ∗ Cv via the solution to an adjoint problem for any v, we applyalgorithm[81℄ toestimate the eigenvalues of C ∗ C

forthe problems, theirnite approximationsand v results, also

some standard algorithmslikethe gradientmethod, algorithm, are

presented there We note that the solution to the hlet problem (0.8)(0.10) or

to the Neumann problem (0.8), (0.9), (0.11) is understood in the weak sense, the nite

methodfor them is and the proof of v of the method is

not trivial

problem(0.8)(0.10)fromtheobservationatthenaltimemoment: Cu := u(x, T ) = ξ(x).This problem is well known under the name parab equations b d in time h

has many in and up tonow therehave been manypapers devoted to

it(see [4,5,6, 8, 12,18,24, 31,45,68℄and the therein) However, amongthese

works onlya few are devoted tothe oftime-dependent ts[1, 25,31,45, 53℄

Parab equations kward intimeareseverelyill-posedasthefollowingsimpleexample

The problemis to the initial v from u(x, 1) = ξ(x)

UsingFourier expansion of v, wehave the representation

∞

X

n=1

inthe initial

temperature

Now we return to the problem of the initial v in the hletproblem(0.8)(0.10)fromtheobservationatthenaltimemoment: Cu := u(x, T ) = ξ(x)

Wesometimeswriteu(x, t; v)oru(v)insteadofu(x, t)toemphasizethedep ofuon

v Followingthegeneral hstatedabo eto vweminimizethe

J 0 (v) := 1

2 ku(·, T ; v) − ξk

2

with resp to v ∈ L 2 (Ω) We willsee that the operator v → Cu(v) : L 2 (Ω) → L 2 (Ω) is

the problemofsolving the equation Cu(v) = ξ is ill-posed It follows thattheabo eminimizationproblemisalsoill-posed TostabilizeitweminimizetheTikhonov

p(x, t) = 0 onS.

The optimizationproblemis then bythe nite methodin

vari-ables and it isproved that the solution of the optimization problem verges

(weaklyorstronglydependsonthesmoothnessofthedata)tothesolutionofthe tinuous

one Wefurther itin time usingthe splitting methodto split multi-dimensional

problems into a of one-dimensional ones We derive a formula for the gradient

of the fully via an adjoint problem and then apply the

gradient method to solve it n The algorithm is then tested on several b

h-mark examples toshow the of our h We alsoapply algorithm

toestimatethe singularvaluesof the version oftheoperatorC Here, asabo e,

C is the linear operator from v into Cv := Cu(v) − C u o with

Let us observations (0.22) First, any measurement is an a eraged pro that

is of the form (0.22) h a kind of observations is a generalization of point

observations Indeed, letx i ∈ Ω, i = 1, , N be given, S i be a neighbourhood of x i

point does not always have a meaning, but it does in the abo e a eraged sense Third,

with this kind of observations, the data need not be always a ailable at the whole

domain and at any time Thus, our problem setting is new and more than the

related ones, where one requires either 1) the knowledge of u(x, T ) in the whole spatialdomainΩthat ishardly realizedin (see the tsurvey [12℄,or[25, 61℄ andthe

therein), 2) orthe measurements of u in ω × (τ, T ), where ω is asubdomain of

ourevolutionsystemisgeneratedbytheNeumannproblem(0.8),(0.9),(0.11) Theinverse

problem we study isto the initial v in(0.9), when the solution u isgiven on a part of the boundary S Namely, let Γ ⊂ ∂Ω and denote Σ = Γ × (0, T ).Our aimisto the initial v fromthe measurement ϕ ofthesolution uon Σ:

parab equations [24, 31, 45, 58℄ tly, Klibanov proved some stability estimates

for this inverse problem [38, 39℄ Unfortunately, up to now there are very few studies on

n methodsforthis problem The workby hëv[13℄ et alisthe only

onthis asp h wehavefound Thus, the solutionmethodforthis problemproposed

in this hapter is a new tribution to eld We note that this problem has the root in

theinverseheat problem(IHCP)whereonedeterminesthe temperature

and heat uxonan part of the boundary fromthe temperature andthe

heat ux on the part of it (see [4, 9, 24, 45℄) The t formulation

for IHCP requires the initial given [4, 9℄ However, as IHCP be regarded

needed [24, 45, 58℄ The uniqueness with stability estimates are proved in(see [24, 31,45,

58℄) Using the same method presented in Chapters 2 and 3, we minimize the Tikhonov

For we for the Ω is an open parallelepiped in R n , n = 1, 2, 3

one- and two-dimensional variables only

Parts of the thesis havebeen published in

1 Nguyen Thi Ngo Oanh, A splitting method for a kward parab equation with

time-dependent ts, Computers & with ations 65(2013),

1728 (Chapter 2)

2 Dinh Nho Hào and Nguyen Thi Ngo Oanh, Determination of the initial

in parab equations from integral observations, Inverse Problems in e and

Engineering(to appear) doi: 10.1080/17415977.2016.1229778 (Chapter 3)

3 Dinh Nho Hào and Nguyen Thi Ngo Oanh, Determination of the initial

in parab equations fromboundary observations, Journal of Inverse and Ill-Posed

Problems 24(2016), no 2, 195220 (Chapters 1and 4)

and have been presented at

1 8thInternational "InverseProblems: Modeling&Simulation",2328May,

2016, Ölüdeniz, Fethiye, Turkey;

2 Mini-workshop on "Analysis and of PDEs", 29June 2016, Vietnam

In-stitute for Adv Study in

3 Vietnam-Korea workshop on in 2024 February, 2017,

Da Nang, Vietnam;

Auxiliary results

In this hapter, we intro some notions in Sobolev and present

well-posedness results relatedtothe hlet and Neumann problems for parab equations

The main part ofthis hapteris devoted tothe nite methodin variables

and its v for the hlet and Neumann problems This v of nite

heme is a new resultfor the weak solution The splittingmethodissuggested

andprovedtobestable Avariationalproblemandits versionarepresentedand

anew v resultforweaksolutionisproved algorithmforapproximating

eigenvalues of a matrix ispresented

where ν isthe outer unit normal v to S.

Whenthe tsof (1.7),v, g andf aregiven, theproblemof uniquelysolvingu(x, t)

fromthe system (1.7)(1.9) or(1.7), (1.8), (1.10) is the dire problem[24, 82℄

To study these problems, we intro the followingstandard Sobolev (see [24,29,

Denition 1.1.4 The H 1,1 (Q) is the set of all elements u(x, t) ∈ L 2 (Q) havinggeneralized derivatives

Denition1.1.5 The H 0 1,0 (Q)isthe set ofallelementsu(x, t) ∈ H 1,0 (Q)vanishing

kuk 2 L 2 (0,T ;B) =

Z T 0

Thesolutionsofthe hletproblem(1.7)-(1.9),andtheNeumannproblem(1.7),(1.8),(1.10)

are understood in the weak sense asfollows:

Denition 1.1.7 AweaksolutioninW (0, T ; H 1

0 (Ω))of theproblem(1.7)-(1.9)isation u(x, t) ∈ W (0, T ; H 1

0 (Ω)) satisfyingthe identity

Denition 1.1.8 A weak solution in W (0, T ; H 1 (Ω)) of the problem (1.7), (1.8), (1.10)

is a u(x, t) ∈ W (0, T ; H 1 (Ω)) satisfyingthe identity

Due to [24, p 3546℄, [41℄, [82, p 141152℄ and [83, Chapter IV℄ we have the following

results about the well-posedness of the hlet and Neumann problems

Theorem1.1.1 Letthe onditions (1.1)(1.6) b satised Thefollowingstatementshold:

1) There exists a unique solution u ∈ W (0, T ; H 1

0 (Ω), a ij , b ∈ C 1 ([0, T ]; L ∞ (Ω)), i, j = 1, , n and there exists a onstant

µ 1 that |∂a ij /∂t|, |∂b/∂t| ≤ µ 1, then u ∈ H 0 1,1 (Q)

1)Thereexistsaunique solutionu ∈ W (0, T ; H 1 (Ω))totheNeumannproblem (1.7),(1.8),(1.10) Furthermore,there existsapositive onstantc N independentof theinitial ondition

v, theboundary ondition g and the right-hand sidef (only depends on a ij, b andΩ)that

2) If v ∈ H 1 (Ω), g ∈ H 0,1 (S), a ij , b ∈ C 1 ([0, T ]; L ∞ (Ω)), i, j = 1, , n and there exists

a onstantµ 1 that|∂a ij /∂t|, |∂b/∂t| ≤ µ 1 thenu ∈ H 0 1,1 (Q) Inthis ase,there exists

a onstant, denoted again by c N that

kuk H 1,1 (Q) ≤ c N 

kf k L 2 (Q) + kgk H 0,1 (S) + kvk H 1 (Ω)



In additiontoDenition 1.1.7and Denition 1.1.8, weintro the following denitions

The weaksolutioninH 1,0 (Q)tothe hletproblem(1.7)(1.9)and theNeumannlem(1.7), (1.8), (1.10) are understood inthe weak sense as follows:

prob-Denition 1.1.9 A weak solution in H 1,0 (Q) to the problem (1.7)(1.9) is a

u ∈ H 0 1,0 (Q)satises the identity

Denition 1.1.10 A weak solution in H 1,0 (Q) to the problem (1.7), (1.8), (1.10) is a

u ∈ H 1,0 (Q)satises the identity

To study the variational problemfor data assimilationin heat we need some

results related tothe adjoint problems and Green's formula The following results be

proved in the same wa as in[82, Ÿ3.6.1.,p 156158℄

By hanging the time and using the result of Theorem 1.1.1, we see that there

exists a unique solution p ∈ W (0, T ; H 1

0 (Ω)) and p also satises an a priori inequalitysimilar to(1.15) We havethe following result:

Theorem 1.2.1 Suppose that the onditions (1.1)(1.4) hold Let y ∈ W (0, T ; H 1

y = 0 on S, y(x, 0) = b Ω in Ω,

(1.21)

with b Q ∈ L 2 (Q), and b Ω ∈ L 2 (Ω) Assume that a Q ∈ L 2 (Q), a Ω ∈ L 2 (Ω) and p ∈

W (0, T ; H 0 1 (Ω)) is the weak solution to the adjoint problem (1.20) Then we have Green'sformula

∂ N p = a S onS, p(x, T ) = a Ω inΩ,

(1.23)

where a Q ∈ L 2 (Q), a S ∈ L 2 (S), and a Ω ∈ L 2 (Ω) We dene the solution to this problem

by a p ∈ W (0, T ; H 1 (Ω)) satisfyingthe variationalproblem

We alsoprove that there exists aunique solution p ∈ W (0, T ; H 1 (Ω)) tothis problemand it satisesan apriori inequality similarto (1.16).

Theorem 1.2.2 Let y ∈ W (0, T ; H 1 (Ω)) b the solution to the problem

∂ N y = b S on S, y(x, 0) = b Ω in Ω,

(1.24)

with b Q ∈ L 2 (Q), b S ∈ L 2 (S), and b Ω ∈ L 2 (Ω) Suppose that a Q ∈ L 2 (Q), a S ∈ L 2 (S),

a Ω ∈ L 2 (Ω) and p ∈ W (0, T ; H 1 (Ω)) is the weak solution to the adjoint problem (1.23).Then we have Green's formula

In the next two we present the nite method for solving the

problems Note that the solutions to the hlet and Neumann problems studied in

this thesis are understood in the weak sense, the v results of the nite

method for them are not trivial For y of presentation, we willseparately

e thenite methodfor one-dimensionaland multi-dimensionalproblems

Inthis weintro thenite methodtoapproximateweak solutionsto

the one-dimensional problems by using method

Let Ω = (0, L) and Q = (0, L) × (0, T ), S = {0, 1} × (0, T ) We subdivide the interval

(0, L) into N x subintervalsby the uniform grid

u(0, t) = u(L, t) = 0 in(0, T ].

(1.26)

The solution to(1.26) isa u(x, t) ∈ W (0, T ; H 1

∂x

∂η

∂x + b(x, t)uη − f η

 dxdt = 0,

∀η ∈ L 2 (0, T ; H 1

0 (Ω)), u| t=0 = v in Ω.

Here, we suppose that g(0, ) and g(L, ) are inL 2 (0, T )

The weak solutionto the problem (1.28) is a u(x, t) ∈ W (0, T ; H 1 (Ω)) satisfyingthe identity

∂x

∂η

∂x + b(x, t)uη

 dxdt

g(0, t)η(0, t)dt +

Z T 0

g(L, t)η(L, t)dt, ∀η ∈ L 2 (0, T ; H 1 (Ω)), u| t=0 = v inΩ.

Putting the approximations in(1.30)(1.33) into(1.27), we obtain

− + a i ) TherighthandsideF (t) = {f ¯ i (t), i = 0, 1, , N x }

b The Neumann problem (1.28)

Similarly tothe hlet problem,puttingthe approximations(1.30)(1.33) in(1.29), we

are given by formulas (1.34) and (1.36)

Thus, we get the following system

(1.40)

The positivesemi-deniteness of Λ in(1.37) and (1.40) is proved asfollows.

Lemma 1.3.1 With e t, the o matrix Λ dened in the system (1.37) and(1.40) is positive semi-denite

Nowwe (1.37)and(1.40)intimeby method Wesubdividethe

interval (0, T ) intoN t uniform subintervals by the grid 0 = t 0 < t 1 < · · · < t N t = T with

t m+1 − t m = ∆t = T /N t, m being time index Denoting u m = ¯ u(t m ), Λ m = Λ(t m ), F m =

where E isthe identity matrix

Denote by (·, ·)and k · k the pro and norminthe R N x

, resp

tively We havethe following result onthe stability of the nite heme

Lemma 1.3.2 The (1.43) is stable

Proof It follows from the rst equation of (1.43) that

On theotherhand, Λ m

ispositivesemi-denite,itfollowsfromKellogg'sLemma[56,

= sup

φ

(φ, φ) ((φ, φ) + ∆t(Λφ, φ) + ∆t 2

Consequently, the nite heme (1.43) isstable

prob-lems

We apply the splitting nite method to the multi-dimensional (n =

2, 3) problems (1.7)(1.9) or (1.7), (1.8), (1.10) The idea of splitting hemes is

to approximate a problem by a of simpler ones The main advantages

of the splitting hemes are: (i) they are stable regardless of the of the spatial

and temporal grid sizes and (ii) the resulting linear systems be easily solved

they are triangular systems Using the hniques presented in [40, 54, 56, 84℄ (see also

[26, 27, 29, 61,62, 76,77℄),we propose the nite heme based onthe formulas

(1.11) and (1.13) for the denition of the weak solutions First, we the problem

in variablesand obtain system ofordinary dierentialequationswith resp tothe

time variable t, then we the obtained system intime by the splittingmethod

We ourself to the Ω is an open parallelepiped inR n , n = 2, 3 Furthermore,

we donot mixed derivativesinthe equation(1.7), althoughin this

betreatinthesamemanner[84℄ Thus, for theeaseof notation,weset a ij = 0 ifi 6= j

and denote a ii = a i , i = 1, , n.

u = 0 on S.

(1.46)

The weak solutionto the problem (1.46) is a u(x, t) ∈ W (0, T ; H 1

0 (Ω)) satisfyingthe identity

n )is the grid point;

• h := (h 1 , , h n ) is the v of spatialgrid size;

• ∆h := h 1 · · · h n;

• e i, i = 1, , n being the unit v in the x i i.e e 1 = (1, 0, , 0), , e n = (0, , 0, 1).

Around hgrid point,we dene the followingsubsets of Ω:

Ω h := {k = (k 1 , , k n ) : 1 ≤ k i ≤ N i − 1, ∀i = 1, , n}. (1.54)The set of the of all boundary grid points isdenoted by Π h, that is

u(k, t) its approximate value at (x k , t) Suppose that u = {u ¯ k , k ∈ ¯ Ω h } is a grid

dened in Q hT := ¯ Ω h × (0, T ) h have the rst weak derivative with resp to t Wedene

When u h is a grid we denote u hx i by u x i.

For agrid u ¯ we denethe following interpolationsinQ:

˜¯u(x, t) := ¯u k (t), (x, t) ∈ ω(k) × (0, T ). (1.61)2) Multi-linear:

Lemma 1.4.3 Suppose that the hypothesis of Lemma 1.4.1 isfullled Thenif {ˆ¯ u(x, t)} h

weakly onvergestoa u(x, t)inL 2 (Q)asthegridsizehtendstozero, these e

{˜¯ u(x, t)} h also weakly onverges to u(x, t) in L 2 (Q) Moreover, if {ˆ¯ u | S } h weakly onverges

to u | S in L 2 (S) then {˜¯ u | S } h alsoweakly onverges to u | S in L 2 (S)

Lemma 1.4.4 Suppose that the hypothesis of Lemma 1.4.2 isfullled Thenif {ˆ¯ u(x, t)} h

strongly onverges to a u(x, t) in L 2 (Q) as the grid size h tends to zero, the

se-e{˜¯ u(x, t)} h alsostrongly onvergestou(x, t)in L 2 (Q) Moreover,if{ˆ¯ u | S } h onverges

to u | S in L 2 (S) then {˜¯ u | S } h also onverges to u | S in L 2 (S)

Lemma 1.4.5 Suppose that the hypothesis of Lemma 1.4.2 isfullled and i ∈ {1, , n}.Then if the se e of derivatives {ˆ¯ u x i (x, t)} h weakly onverges to a u(x, t) in

L 2 (Q)as htends tozero, the se e{˜¯ u x i (x, t)} h also onvergesto u(x, t)in L 2 (Q)where

a The hlet problem (1.46)

Substituting the integrals (1.68)(1.71) into the rst equation of (1.47), we have the

fol-lowing equality (dropping the time variable t for a moment)

∆h

Z T 0

Using the summation by parts formula together with the u ¯ k = ¯ η k = 0 when

− ¯a k i u ¯

h 2 i



¯ k ,

(1.74)

with ¯a k

into (1.73) and approximating the initial u ¯ k (0) =

(1.75)

withu = {u ¯ k , k ∈ ¯ Ω h },the v = {v ¯ k , k ∈ ¯ Ω h }beingagrid approximatingthe initial v and F = {f ¯ k , k ∈ Ω h } where ¯ k

is dened in formula (1.67) The

ts matrix Λ i has the form

h 2 i

¯

u k − ¯ u k−e i

− ¯a

k i

h 2 i

¯

u k+e i − ¯ u k

, 2 ≤ k i ≤ N i − 2,

¯a k i−

h 2 i

¯

u k − ¯a

k i

h 2 i

¯

u k+e i − ¯ u k

, k i = 1,

¯a k i−

h 2 i

¯

u k − ¯ u k−e i

+ ¯a

k i

h 2 i

the followingpart

b The Neumann problem (1.48)

Putting (1.68)(1.72) in the rst equation of (1.49), for the ease of notation, we drop the

time variable t for a moment,we have

arbi-traryand xed k 2 , , k n, usingthe summation by parts, weobtain

¯ k − ¯a k 1 u ¯

h 2 1

¯ k r ,

(1.78)

with ¯a k

1 The similar terms in the x 2 , , x n are treated in the same

wa Then, this equality into (1.77) and approximating the initial in

the equation of (1.49), we obtain the following system approximating the original

h 2 i

¯

u k − ¯ u k−e i

− ¯a

k i

h 2 i

h 2 i

Proof Withoutloss of generality,we assume n = 2and i = 1 We have

N X 1 −1

¯a (k 1 ,k 2 ) 1



U (k 1 ,k 2 ) − U (k 1 +1,k 2 )  2

+ 1 2

1 is positive semi-denite The proof is

Next, we prove the boundednessof the solution of (1.75) and (1.79)

Lemma 1.4.7 Let u ¯ b a solution of the problem (1.75)

a) If v ∈ L 2 (Ω), then exists a onstant c independent of h and the o of theequation that

Multiplyingthe both sides ofthe equality (1.85)by 2,applying hy's inequalitytothe

rst term in the righthand side, noting that b k ≥ 0, weobtain

≤ c 

∆h

Z T 0