Error analysis for incompressible viscous flow based on a sequential regularization formulation

In this dissertation we will discuss error analysis of a sequential regularizationmethod SRM to solve time dependent incompressible Navier-Stokes equations.From both theoretical and nume

ERROR ANALYSIS FOR INCOMPRESSIBLE

VISCOUS FLOW BASED ON

A SEQUENTIAL REGULARIZATION FORMULATION

LU XILIANG

NATIONAL UNIVERSITY OF SINGAPORE

2006

VISCOUS FLOW BASED ON

A SEQUENTIAL REGULARIZATION FORMULATION

LU XILIANG

(M.Sc National University of Singapore)

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICSNATIONAL UNIVERSITY OF SINGAPORE

2006

Prof Lin Ping has served as my advisor since a meeting in July 2000, in my morethan five years in the Department of Mathematics at National University of Singa-pore, Prof Lin committed significant time and effort in support of my completion

of the research I am most grateful to have Prof Lin as my supervisor, and I dohope our working relationship may continue in the future

At the same time, I would like to thank Prof Liu Jian-Guo for his kind help, I wouldlike to thank Prof Olivier Pironneau, Prof Fr´ed´eric Hecht and Prof Antoine LeHyaric, who made the free software FreeFEM++ which saves me a lot of time onwriting code In addition, I would like to express my thanks to my colleagues andfriends, Dr Tang Hongyan, Dr Zhang Ying, Dr Liang Kewei, Pan Suqi and JiaShuo who gave me a lot of help and encouragement in my research and my life

I would like to thank National University of Singapore for awarding me the ResearchScholarship and providing me with a helpful research environment

Finally I would like to dedicate this dissertation to my parents

i

4.1 A Priori Estimations for Semi-Discrete Equations 284.2 Existence and Uniqueness of the Solution 33

5.1 First Order Schemes 425.2 Crank-Nicolson Scheme 48

6.1 Error Estimation to a Spatially Discrete Scheme 556.2 Error Estimation to Full Discrete Scheme 63

ii

7.1 Energy Estimations 717.2 Finite Element Estimation 74

8.1 Flow with Exact Expression 818.2 Cavity Flow 878.3 Flow past a circular cylinder 95

In this dissertation we will discuss error analysis of a sequential regularizationmethod (SRM) to solve time dependent incompressible Navier-Stokes equations.From both theoretical and numerical point of view, the most difficult part to solveNavier-Stokes equations is how to deal with the divergence free condition Thissequential regularization formulation can treat this difficulty efficiently.

We review a few existing numerical methods for Navier-Stokes equations in chapter

1, especially projection method and penalty method We then introduce the tial regularization method, including the derivation and various formulations of thismethod

sequen-Chapter 2 is a preliminary part We collect a number of well-known inequalitieswhich will be used in our analysis in later chapters In chapter 3, we prove a fewlemmas for equations related to sequential regularization formulation Chapters 4

to 7 will include our main results The existence and uniqueness of the sequentialregularization solution are proved in chapter 4 In chapter 5, we consider semi-discretization in time We obtain the optimal error estimation for each scheme Inchapter 6, the error estimation of semi-discretization in spatial variables and fully

discrete scheme are obtained We focus on a special case of SRM (α1 = 0) in chapter

7 and give similar error estimation for semi-discretization in spatial variables

We include three numerical examples in chapter 8 The first example has exact

i

iisolution and can be used to verify the convergence The second and third examplesare both well-known problems in computational fluid dynamics We can compareour results with benchmark solutions In the last chapter, we conclude this thesisand point out a few directions which we will work in future.

A derivation will be given in the appendix A Assume Ω is a bounded connected

domain in R n (n = 2, 3) with smooth boundary Let u describe the velocity field

of the fluid, and p describe the pressure, then (u, p) satisfies dimensionless

Re , and Re is Reynolds number.

Although the Navier-Stokes equations were derived in the 19th century, our derstanding of them is still limited, such as that the regularity of the solutions in ageneral three dimensional domain is still open, which is one of ”the Millennium Prob-

un-1

Chapter 1 Introduction 2lems” Numerical simulation of the Navier-Stokes equations also has a long history.

A lot of methods have been proposed The standard discretization which includefinite difference method, finite volume method, conformal finite element method onthe divergence free space, non-conformal finite element method and spectral methodare discussed in [30][11][15][26][17][8] The main difficulty to solve the Navier-Stokes

equations numerically is that the velocity field u and the pressure p are coupled

by incompressible constraint divu = 0 we need to find a proper way to disposethis difficulty when we try to use direct discretization There are some approaches

to overcome this difficulty, such as projection method and penalty method In thefollowing, we will briefly introduce these two methods, and then present sequentialregularization method

The projection method was firstly introduced by Chorin and Temam [10][38]

It is one kind of fractional-step methods We solve an intermediate velocity un+12

regardless of the divergence free condition, then project it to the divergence freespace by the Helmohltz-Hodge decomposition Hence it decouples the velocity andthe pressure The semi-discrete scheme of the projection method can be written asfollows

both sides of un+1 −u n+ 12

4t + ∇p n+1 = 0, and obtain a Poisson equation for p n+1 with

homogeneous Neumann boundary condition We can then solve p n+1 and update

un+1 by substituting p n+1 into the first equation The scheme (1.4) - (1.5) andits variations have been widely used to find numerical solution of Navier-Stokesequations The convergence has been analyzed as well, see [31][32][34] We will give

a few remarks on the projection method, especially for scheme (1.4) - (1.5)

Remark 1.1 1 In the equations (1.4), to calculate the intermediate velocity, there

are different ways to approximate the nonlinear convection term One simple

artificial stabilizer is not required, but the resulting stiffness matrix is not ric.

it is not necessary to satisfy the homogeneous Dirichlet boundary condition (we use

boundary condition, but we do not have any explicit boundary condition for pressure

in Navier-Stokes equations, which also prevents us from obtaining accurate imation for the pressure near the boundary.

approx-Besides the projection method, reformulation methods are also widely used toovercome the difficulty of divergence free condition The idea is to reformulate the

original system by adding a term involving a small parameter ² When ² tends to

0, the solution of the reformulated system will approach the exact solution Thepenalty method, artificial incompressibility method and the sequential regularizationmethod belong in this category

Penalty Method

We have the expression p ² = −1

²divu² from the second equation, then substitute

it to the first equation to obtain an equation which only involves u² In [5][33], thereare more details of the penalty method and its error analysis

In [3], an iterative reformulation method is introduced to solve the algebraic equations (DAE) It is so-called sequential regularization method (SRM).The DAE is an ordinary differential equation coupled with an algebraic equation.Partial differential algebraic equations (PDAE) can be viewed as an extension ofDAE The Navier-Stokes equations are important examples of PDAE In [23], how

differential-to apply SRM differential-to the Navier-Sdifferential-tokes equations was considered This thesis is acontinuation of the work in that paper

When we consider a DAE (or PDAE) of form

xt + Ax + By + q = 0,

Cx + r = 0.

(1.9)

Where A, B, C can be matrixes (DAE) or differential operators (PDAE) Since

equation (1.9) is an index 2 DAE, it is ill-posed in a certain sense, and directdiscretization does not work well Regularization methods such as penalty methodreformulate the second equation by adding a small perturbation term which involves

y, to reduce the index of system Then we can apply straightforward discretization

to the perturbed system We will focus our attention on the Navier-Stokes equations(1.1) - (1.3) in the next section

The Navier-Stokes equations are ill-posed for pressure in the following sense If

we have small perturbation for divergence of u, say divu = δ, then the change of

terminology, Navier-Stokes equations are so-called index-2 PDAE (cf [6][23][27])

On the other hand, if the initial condition u(0) is divergence free, then we have a lot

of formulations mathematically equivalent to divu = 0 If we use divut= 0 instead

of divu = 0, it is equivalent to take divergence of equation (1.1) We thus obtain a called pressure-Poisson equation But there is a weak instability associated with thisformulation, i.e given small perturbation for the second equation (divut = δ), then divu = δt will increase when time goes on The so-called Baumgart’s stabilization

so-(see [4]) can be used here to enforce the stability To do so, we rewrite the constraintdivu = 0 to its equivalent form divut + αdivu = 0, where α is a positive number,

then divu will exponentially decay when time increases However, we still need tosolve the Poisson equation for pressure, and thus to impose an artificial boundary

condition since there is no boundary condition for p in the Navier-Stokes equations.

As we mentioned before, reformulation methods can be applied to Navier-Stokes

equations For instance, if we modify the equation divu = 0 as divu + ²p = 0,

then we have the penalized formulation The penalty method is well-posed, but in

practice, we need to choose very small ² to reduce the reformulation error That

could make the system stiff When we use explicit time discretization, the time step

4t would be too small to be used for long time computation The penalty method

also causes an initial layer for the pressure To see this, consider divu + ²p = 0 at initial time t = 0 We obtain p(0) = 0 since divu(0) = 0, but the exact pressure for

Navier-Stokes equations may be nonzero at the initial time This layer will disappear

after an O(²) time To avoid initial layer, we can combine the penalty method

with Baumgart’s stabilization, that is, reformulate the incompressible condition as(divut +αdivu)+²p = 0 This is the idea of sequential regularization method (SRM)

Chapter 1 Introduction 6developed in [23].

The SRM is an iterative penalty method for the Baumgart stabilized formulation

Let two constants α1 ≥ 0 and α2 > 0 We replace the incompressible condition by

α1divut + α2divu = 0 and modify it as α1divut + α2divu = ²(p0− p), where p0 is a

given function Hence, for any initial guess p0, we can define a sequence of functions(us , p s) which satisfy the following system:

(α1div(us)t + α2divus ) = ²(p s−1 − p s ), (1.11)

where s = 1, 2, 3 · · · For any given s, p s−1 is a known function which is obtained

from the previous iteration We obtain an explicit expression of p s in term of us and

p s−1 from equation (1.11), then we substitute p s to (1.10) This eliminates p s fromthe coupled system and we obtain a partial differential equation which only involvesunknown function us We can then solve this PDE to obtain us , and recover p s from(1.11) This is the iterative procedure of SRM

To avoid technique difficulty from nonlinear convection term, we only consider

2-dimensional Navier-Stokes equations in a fixed time interval (0, T ) during the thesis.

We believe the results can be extended to long time case and 3D case From thefollowing convergence of SRM

² can be chosen as a reasonable small number, the number of iterations s will reduce

reformulation error dramatically At the same time, we can see divusis exponentiallydecreasing from (1.11), this can be demonstrated by the numerical examples inchapter 8

We first describe some notations and assumptions For scalar functions we use L p(Ω)

to denote the space of functions which are p th-power integrable in Ω and

If the domain Ω is fixed, we can simply denote L p (Ω) as L p The most important

case is p = 2 We define the inner product (·, ·) in L2 by (u,v)=RΩuvdx, and let

0≤|α|≤m

kD α f k2)12.

For simplicity, we denote k · k m = k · k H m , m is a positive integer When m = −1,

H −1 is defined as the dual space of H1

0 In general, H m is defined by Fouriertransformation or interpolation process (see [1][12])

For vector functions, we use the similar notations but with bold characters such

as Lp, Hm etc A vector function in Hm implies every component of the function

7

Chapter 2 Preliminary 8

is a member of H m Other vector function spaces are defined in the same way

Throughout the thesis, we always use u, v, f · · · to describe scalar functions and

u, v, f · · · to describe vector functions, and the domain will be fixed as Ω without

specific illumination

We list some widely used inequalities here You can find these inequalities in

many analysis textbooks (see [12]) We define C as a genetic constant which does

not depend on the choice of functions

kuk L4 ≤ Ckuk12kuk11,

where the domain is in R2

• Young inequality:

ab ≤ ²a p + c ² b q ,

where 1 < p < ∞, ² > 0, 1

p +1

q = 1, c ² is a constant which only depends on ².

• Gronwall inequality in differential form:

Let y(t) be a nonnegative, absolutely continuous function in [0, t] and satisfy

for almost every t, the differential inequality:

y 0 (t) ≤ a(t)y(t) + b(t), where a(t) and b(t) are nonnegative, summable functions in [0, t] Then we

• Discrete Gronwall inequality:

Let y n , a n , b n and c n be nonnegative sequences, satisfying

We will consider the time dependent incompressible Navier-Stokes equations

(1.1) - (1.2) with homogenous Dirichlet boundary condition u| ∂Ω = 0 and initialvalue u(0) = u0 To deal with the nonlinear convection term in Navier-Stokesequations, we introduce two operators as follows (see [36][37])

B(u, v) = (u · ∇)v,

¯

2(divu)v.

Chapter 2 Preliminary 10

For Navier-Stokes equations, we can replace B(u, u) by ¯ B(u, u) since divu = 0 Let

¯b(u, v, v) = 0, ∀u, v ∈ H1

For the trilinear form b, we can prove the following inequalities by combination of

integration of parts, H¨older’s inequality and Sobolev inequality (see for instance[37])

Lemma 2.1 Assume Ω ∈ R n , then the trilinear form b is defined as bounded linear

From the equation (2.1), we immediately have that the lemma 2.1 and 2.2 are

also true for trilinear form ¯b For simplicity, we focus on ¯ B and ¯b in the thesis, but

all results we obtain later are valid for B and b as well.

Chapter 3

Some Estimations to SRM

Equations

we recall that the sequential regularization reformulation of Navier-Stokes equations

reads: given initial pressure p0, for s = 1, 2, 3 · · · , solve (u s , p s) from the followingsystem,

α1, to rewrite the equations to a canonical form From now

on, we assume α1 = 1, α2 = α, and consider the case of α1 = 0 in chapter 7

Substituting p s = p s−1 − div(α1(us)t + α2us ) to (3.1) and eliminating p s, we have

an equation which only involves us Since at each step of iteration, we have very

similar equation Then we can omit the iteration index s and rename the right hand

side as f to get the following equation:

12

ut − 1

This PDE is an implicit parabolic equation and −∇div is a degenerate elliptic

operator, the existence of solution is not trivial We will prove its existence in nextchapter In this chapter, we assume existence of the solution and establish someenergy estimations for the equations (3.4) - (3.5)

This chapter will be organized as follows In the first section we will obtain a fewestimations for (3.4) - (3.5) Most of them will be used in later chapters, to obtainthe error estimation for the discrete system At second part of this chapter, we willprove the reformulation error of SRM by using results in the first section

Lemma 3.1 Define operator Au = −1

The constant C here does not depend on the choice of ² and u.

Proof: Let w = Au, p = −1

²div(ut + αu) and g = divu Firstly we solve g from

Chapter 3 Some Estimations to SRM Equations 14

u and p satisfy the non-homogenous Stokes equation,

From inequality (3.8), (3.9) and definition of p, we obtain (3.6) and (3.7) 2

Remark 3.1 The generic constant C in the lemma 3.1 is independent of the choice

Proof: Multiply u for both side of equation (3.4), we have,

2+ νk∇uk2 + ¯b(u, u, u) = (f, u).

We notice ¯b(u, u, u) = 0 and (f, u) ≤ ν

2k∇uk2+ C1kfk2, and hence1

Since we only discuss the 2-dimensional case, then

Chapter 3 Some Estimations to SRM Equations 16

If we assume more regularity on f and u(0), we can remove the integral R0T inthe estimate (3.10)

u be the solution of equations (3.4), (3.5) Then we have:

Sup t∈[0,T ]

Let t = 0 at above equality, the Young inequality yields

Multiplying ut at both sides of above equation yields

Chapter 3 Some Estimations to SRM Equations 18

Then multiplying Au at both sides of equation (3.4), we have

Finally applying lemma 3.1, we obtain the estimation of each term in (3.14) 2

For lemma 3.1, we can relax the divergence free condition for the initial value

u0, it will be useful when we consider higher order regularity

Lemma 3.2 Define operator Au = −1

Proof: The proof is almost same as lemma 3.1 Let w = Au, p = −1

²div(ut + αu) and g = divu Firstly we solve g from the ODE

2+ ku tt k2)dt, we will have two inequalities at following The first version

requires a global compatibility Taking the time derivative for equation (3.4), wehave

utt − 1

Chapter 3 Some Estimations to SRM Equations 20

Theorem 3.3 Assume all conditions in Theorem 3.2 Moreover, f t ∈ L2(L2), g = f(0) + ν∆u0− ¯ B(u0, u0) ∈ H1

0 Multiplying Au t at both sides of equation (3.25), we have

(utt , Au t ) + kAu t k2+ ¯b(u t , u, Au t ) + ¯b(u, u t , Au t) = (ft , Au t ).

d

2

1+ kAu t k2 ≤ M3(ku t k21+ kf t k2).

Applying Gronwall inequality, we obtain

Multiplying utt at both sides of equation (3.25), we have

ku tt k2+ (Au t , u tt ) + ¯b(u t , u, u tt ) + ¯b(u, u t , u tt) = (ft , u tt ).

Remark 3.3 1 Consider sequential regularization formulation (3.1) - (3.3) (let

α1 = 1 and α2 = α), if the following over-determined Neumann problem

∆q = div(f(0) − ¯ B(u0, u0)) in Ω, (3.29)

∇q| ∂Ω = ∆u0+ f(0) − ¯ B(u0, u0)| ∂Ω (3.30)

the systems as:

Chapter 3 Some Estimations to SRM Equations 22

2 Since the equations (3.29) - (3.30) are over-determined, we need the compatibility

non-local and virtually uncheckable for given data In the absence of such compatibility condition, we will discuss another type of estimation in the follows.

Without assumption of global compatibility, the regularity maybe breakdownwhen time is near 0, we have the following estimation

Theorem 3.4 Assume all the conditions as in theorem 3.2, moreover f t ∈ L2(L2),

Then multiplying utt at both sides of equation (3.25), we have

ku tt k2+ (Au t , u tt ) + ¯b(u t , u, u tt ) + ¯b(u, u t , u tt) = (ft , u tt ). (3.32)

Multiplying t at both sides of equation and integrating from 0 to t, noting

|t¯b(u t , u, u tt )| ≤ C4tku t k1kuk2ku tt k

Combining all above inequalities together we obtain this theorem 2

Corollary 3.1 Fix t0 ∈ (0, T ), ∀t ≥ t0, with same assumption as theorem 3.4, we have

Proof: From theorem 3.2, we know k∇divu t (t0)k ≤ C², then we apply lemma 3.2

by replacing the initial time 0 by t0, we obtain ∀t > t0,

Then the conclusion is straightforward from theorem 3.4 2

Remark 3.4 We can compare theorem 3.3 and theorem 3.4 They both have the

stronger assumption in the theorem 3.3, we obtain a stronger conclusion, that is, the energy is finite up to t = 0 Without global compatibility assumption in theorem

esti-mation will be the same as theorem 3.3 As we mentioned before, when we discuss the discrete system, energy estimation for high order derivative is needed We will mainly use theorem 3.3 in this thesis See some remarks for results related to using

Chapter 3 Some Estimations to SRM Equations 24

to theorem 3.4 will be established and be essential for finite element analysis.

In this section, we will give the convergence result for the sequential regularizationmethod The proof by using the technique of asymptotic expansion is given in [23].Here is another simplified version of proof by using the idea in [24] Let (us , p s) be

the solution of SRM equations (3.1) - (3.3), where α1 = 1 and α2 = α Eliminating

p s from equations (3.1) - (3.3), we obtain

(us)t − 1

Define u to be the solution of Navier-Stokes equations (1.1) - (1.3) We notice thatNavier-Stokes equations are equivalent to following equations:

For the linear auxiliary equation

we have following estimation.

Lemma 3.4 Let function V, W satisfy

Proof: It is nothing but theorem 3.1 2

Using above lemmas, we have estimation for es Suppose our initial guess p0 isnot too large (R0T kp0k2

then ∀² ≤ ²0, we have following theorem

Theorem 3.5 Let e s = u − u s , h s = p − p s , then we have

Chapter 3 Some Estimations to SRM Equations 26

Proof: Subtracting equation (3.35) - (3.37)from equation (3.1) - (3.3) (α1 = 1,

Step 2: Assume s = k is true, let s = k + 1.

Claim (3.48) and (3.47) (for s = k) implies

Chapter 4

Existence and Uniqueness

As we mentioned before, the typical equations when we apply SRM to Navire-Stokesequations are equations (3.4) - (3.5) We have obtained a few estimations in lastchapter if we assume that the solutions exist In this chapter, we will establish theexistence and uniqueness of the solution of equations (3.4), (3.5) The idea is quitestandard We first establish the similar energy estimations for the time discretized

equations Then when the time step 4t approaches to 0, the limit function is the

strong solution of equations (3.4) - (3.5)

4t + αu n+1 ) − ν4u n+1 , n = 1, 2, 3 and u0 = u(0), when

Since u(0) is divergence free, the first term at the right hand side of above equation

is 0 Then we apply Cauchy-Schwarz inequality,

From the inequality (4.7) and the definition of p i, we obtain the first conclusion in

the lemma To obtain the uniform estimation, we take sup in inequality (4.6),

then takeing sup at inequality (4.8) and choosing ² ≤ ²0, we can easily complete the

second part of this lemma 2

Chapter 4 Existence and Uniqueness 30The next step is to consider a semi-discrete scheme in time We use a semi-implicit scheme, i.e an explicit-implicit scheme for the nonlinear convention termand implicit scheme for the remaining The system is:

with homogenous Dirichlet boundary condition Where the u0 = u(0), fn+1 =

f(t n+1) Clearly, the equation (4.10) is a linear second order elliptic equation for any

fixed n The existence and uniqueness is standard, moreover, we have regularity for

equation (4.10)

Lemma 4.2 Assume u(0) ∈ H1

Remark 4.1 Here we take ² as a fixed number In this theorem, we only prove the

in the lemma 4.3 where the bound is independent of ² and 4t.

The existence and regularity for every un is now established In the next part

we will obtain a priori estimate, and pass to limit, to prove the existence of theequations (3.4), (3.5)

Lemma 4.3 For equations (4.10), if we assume 4tPN0 kf n k2

Summing above inequality from 0 to N yields first inequality of this lemma To

finish the second part, taking summation 0 to N − 1 in equality (4.14) We have

Chapter 4 Existence and Uniqueness 32

Proof: In the lemma 4.3, we have proved that sup ku n k and 4tPN1 kuk2

1 are

bounded, we will use these results later Multiplying Au n+1at both sides of equation

(4.10), where the operator Au n+1 is defined in the lemma (4.1), we have

We will define the strong solution as following, the weak solution will be definedlater.

Problem 4.1 (Strong Solution.)

0∩H2),