# Robust preconditioning methods for algebraic problems, arising in multi- phase flow models X

## IT Licentiate theses

## 2011-002

# Robust preconditioning methods for algebraic problems, arising in multiphase flow models

X

IN

H

E

## UPPSALA UNIVERSITY

## Department of Information Technology

## Robust preconditioning methods for algebraic problems, arising in multi-phase flow models

*Xin He*

March 2011

*Division of Scientific Computing*

*Department of Information Technology*

*Uppsala University*

*Box 337*

*SE-751 05 Uppsala*

*Sweden*

http://www.it.uu.se/

Dissertation for the degree of Licentiate of Philosophy in Scientific Computing

ISSN 1404-5117

Printed by the Department of Information Technology, Uppsala University, Sweden

4

**Abstract**

The aim of the project is to construct, analyse and implement fast and reliable numerical solution methods to simulate multi-phase ﬂow, modeled by a coupled system consisting of the time-dependent Cahn-Hilliard and incompressible Navier-Stokes equations with variable viscosity and variable density. This thesis mainly discusses the eﬃcient solution methods for the latter equations aiming at constructing preconditioners, which are numerically and computationally eﬃcient, and robust with respect to various problem, discretization and method parameters.

In this work we start by considering the stationary Navier-Stokes problem with constant viscosity. The system matrix arising from the ﬁnite element discretization of the linearized Navier-Stokes problem is nonsymmetric of saddle point form, and solving systems with it is the inner kernel of the simulations of numerous physical processes, modeled by the Navier-Stokes equations. Aiming at reducing the simulation time, in this thesis we consider iterative solution methods with eﬃcient preconditioners. When discretized with the ﬁnite element method, both the Cahn-Hilliard equations and the stationary Navier-Stokes equations with constant viscosity give raise to linear algebraic systems with nonsymmetric matrices of two-by-two block form. In Paper I we study both problems and apply a common general framework to construct a preconditioner, based on the matrix structure. As a part of the general framework, we use the so-called element-by-element

Schur complement approximation.

The implementation of this approximation is rather cheap. However, the numerical experiments, provided in the paper, show that the preconditioner is not fully robust with respect to the problem and discretization parameters, in this case the viscosity and the mesh size. On the other hand, for not very convection-dominated ﬂows, i.e., when the viscosity is not very small, this approximation does not depend on the mesh size and works eﬃciently. Considering the stationary Navier-

Stokes equations with constant viscosity, aiming at ﬁnding a preconditioner which is fully robust to the problem and discretization parameters, in Paper

II we turn to the so-called augmented Lagrangian (AL) approach, where the linear system is transformed into an equivalent one and then the transformed system is iteratively solved with the AL type preconditioner. The analysis in Paper II focuses on two issues, (1) the inﬂuence of a scalar method pa-

1

rameter (a stabilization constant in the AL method) on the convergence rate of the preconditioned method and (2) the choice of a matrix parameter for the AL method, which involves an approximation of the inverse of the ﬁnite element mass matrix. In Paper III we consider the stationary Navier-

Stokes problem with variable viscosity. We show that the known eﬃcient preconditioning techniques in particular, those for the AL method, derived for constant viscosity, can be straightforwardly applicable also in this case.

One often used technique to solve the incompressible Navier-Stokes problem with variable density is via operator splitting, i.e., decoupling of the solutions for density, velocity and pressure. The operator splitting technique introduces an additional error, namely the splitting error, which should be also considered, together with discretization errors in space and time. Insuring the accuracy of the splitting scheme usually induces additional constrains on the size of the time-step. Aiming at fast numerical simulations and using large time-steps may require to use higher order time-discretization methods.

The latter issue and its impact on the preconditioned iterative solution methods for the arising linear systems are envisioned as possible directions for future research.

When modeling multi-phase ﬂows, the Navier-Stokes equations should be considered in their full complexity, namely, the time-dependence, variable viscosity and variable density formulation. Up to the knowledge of the author, there are not many studies considering all aspects simultaneously.

Issues on this topic, in particular on the construction of eﬃcient preconditioners of the arising matrices need to be further studied.

2

**List of papers**

This thesis is based on the following three papers, which are referred to as

Paper I, Paper II and Paper III.

I. M. Neytcheva, M. Do-Quang and X. He, Element-by-element Schur complement approximations for general nonsymmetric matrices of twoby-two block form. *Springer Lecture Notes in Computer Science (LNCS)*,

5910/2010, 2010.

II. X. He, M. Neytcheva and S. Serra Capizzano, On an augmented

Lagrangian-based preconditioning of Oseen type problems. *Submit-*

*ted to BIT Journal, and under revision now, 2010.*

III. X. He and M. Neytcheva, Preconditioning the incompressible Navier-

Stokes equations with variable viscosity. Technical Report, 2011, Department of Information Technology, Uppsala University, Sweden.

3

4

**Contents**

**2 Coupled model for multi-phase flow**

**11**

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Discretization using the ﬁnite element method (FEM) . . . .

12

Preconditioning technique . . . . . . . . . . . . . . . . . . . .

14

**4 The incompressible Navier-Stokes equations**

**17**

Stationary incompressible N-S equations with constant viscosity 18

Stationary incompressible N-S equations with variable viscosity 23

Incompressible N-S equations with variable density . . . . . .

25

**29**

**31**

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

**33**

**7**

**9**

5

6

**Chapter 1**

**Introduction**

Computational ﬂuid dynamics (CFD) is an important branch of ﬂuid mechanics and computational mathematics. Numerical simulations become more and more irreplaceable and indispensable in modern research, not only because the traditional laboratory experiments are costly, but also because the numerical simulations enable us to model the processes, which cannot be experimentally tested, and extend our capability to reproduce physical phenomena in order to obtain a deeper insight of the underlying processes and their interactions.

Simulation of multi-phase ﬂow is an active research area in CFD and has high impact on numerous applications. For example, the phenomenon of solid-liquid interaction when dropping a solid sphere into a liquid and in particular, the splashing phenomenon, observed when the solid sphere penetrates the liquid’s surface, are very complicated and have been numerically and experimentally studied during the past. Deeper insights and knowledge about this type of multi-phase ﬂow can be obtained through numerical simulations and used in the context of various applications. The author believes that in future we may use the obtained knowledge to help the athletes to control the splash in the diving competition of the Olympic games.

Numerical simulations of multi-phase ﬂow are based on certain systems of partial diﬀerential equations (PDEs), which are, in general, coupled, timedependent and nonlinear. Those PDEs are then discretized, and since we usually aim at obtaining some stationary solutions or at performing fast numerical simulations, implicit time discretization methods are to be recommended. The nonlinearities are handled via some nonlinear methods, such as Newton’s or Picard’s iterations. The computational kernel of these complex numerical simulations is the solution of some linear systems of equations. Since the linear solve is in the most inner loop of long time integration, or a nonlinear solution iteration, or both, it is of importance to use reliable, eﬃcient and fast solution methods for those.

The linear systems, arising in such simulations may have huge dimen-

7

sions, which often outrules direct solution methods due to their high demands for computer resources, enforcing the usage of iterative methods.

Further, the matrices of the linear systems, which are in general nonsymmetric, might be very ill-conditioned, which puts extra demands on the possible preconditioning techniques, which are aimed to be not only numerically and computationally eﬃcient but also robust with respect to various parameters, which arise from the PDEs, the discretization and the solution methods.

In this thesis the multi-phase ﬂow is modeled by a coupled system consisting of the time-dependent Cahn-Hilliard and incompressible Navier-

Stokes equations. How to construct eﬃcient preconditioners for the nonsymmetric matrices arising from the ﬁnite element discretization of the time-dependent and incompressible Navier-Stokes equations with variable viscosity and density is in the focus of this thesis. As to the coupled system, how to construct eﬃcient solution schemes, which permit large time-steps while keeping the accuracy of the obtained solutions, is another main concern in this thesis and the work to follow.

The outline of the thesis is as follows. The coupled model for multi-phase ﬂow is introduced in Chapter 2. The time-dependent Cahn-Hilliard equations and incompressible Navier-Stokes equations are discussed in Chapters

3 and 4. Computational challenges for solving the coupled system are discussed in Chapter 5. A summary of the papers, included in this thesis, is given in Chapter 6 and possible directions of future work are outlined in

Chapter 7.

8

**Chapter 2**

**Coupled model for multi-phase flow**

The phase-ﬁeld model is used to model two immiscible and incompressible ﬂuids. The interface between the ﬂuids is modeled as a narrow interfacial region. The idea that ﬂuid-ﬂuid interface diﬀuses with ﬁnite thickness goes

back to Poisson (1831) and Gibbs. Van der Waals (1893) [57] introduced

the ﬁrst diﬀusive-interface model based on the so-called free energy density and he also suggested that the equilibrium interface proﬁles are those which

case and derived the so-called Cahn-Hilliard equation, which models the creation, movement and dissolution of diﬀusion-controlled phase interfaces.

This diﬀusive-interface model, where the convective eﬀect of the ﬂuid’s motion is not considered, is extensively used in various studies in ﬂuid mechanics, i.e., the study of the dynamics of a near-critical ﬂuid in a shear ﬂow

the interface is measured by the so-called *concentration*, which is also referred to as the *phase field*. The *concentration *takes two distinct values

(for instance +1 and

*−*1) in each of the phases, with a smooth and rapid change between both values in the interface zone. This approach permits to solve the problem by integrating a set of partial diﬀerential equations in the whole domain, without explicit treatment of the boundary conditions at the interface.

By taking into account the convective eﬀect of the ﬂuid’s motion, a convective form of the time-dependent Cahn-Hilliard equation is derived

(see e.g. [19]). More details about the convective Cahn-Hilliard equation

are discussed in Chapter 3. The ﬂuids motion is governed by the Navier-

Stokes equations, therefore the numerical model for resolving the multiphase ﬂow involving incompressible ﬂuids is a coupled system consisting of

9

the time-dependent Cahn-Hilliard (C-H) and incompressible Navier-Stokes

(N-S) equation (see e.g. [23]). More details about the C-H and N-S equations

involved in the coupled system are discussed in Chapters 3 and 4.

For the coupled system, we are in many cases interested in obtaining the stable solutions. Thus, we have to perform time marching in a large time interval. Since we also aim at fast numerical simulations, we would like to use as large time-steps as possible, meanwhile keeping the accuracy of the solution within some prescribed bounds. One possible approach to match the above requirements is to rewrite the linear systems of PDEs arising from the coupled models for the multi-phase ﬂow as diﬀerential-algebraic systems. In this way, the known numerical solution methods for solving the

diﬀerential algebraic equations DAEs (cf, e.g., [48]) can be utilized. The

DAEs formulation permits us to use high-order time integration methods

(of order 4 or higher), and allows large time-steps.

10

**Chapter 3**

**The Cahn-Hilliard equations**

**3.1**

**Formulation**

The formulation of the phase-ﬁeld model is derived via the so-called free energy functional, *F *(*C*), which depends on a variable *C*, referred to as

the *concentration *(the *phase field *). Van der Waals (1893) [57] suggested

that the equilibrium interface proﬁles are the minimizers of the free energy functional, deﬁned as

∫

*F *(*C*) = *f *(*C*(**x***, t*))*d*Ω*,*

Ω where the function *f *(*C*) denotes the free energy density per volume. Here

Ω

*× *(0*, T *] *⊂ *R

*d*

(*d *= 2*, *3) is a bounded, connected domain with boundary

*∂*Ω. Van der Waals derived an explicit expression for the free energy density as *f *= *β*Ψ(*C*) +

1

2

*α*

*| ∇C |*

2

, where *α *and *β *are some constants, proportional to the surface tension coeﬃcient *σ *and the interface width *ϵ*, *α*

*∼ σϵ *and

*β*

*∼ σ/ϵ *(see e.g. [39]). The interaction between the bulk energy, or molar

Gibbs energy, *β*Ψ(*C*), and the interfacial energy

1

2

*α*

*| ∇C |*

2 determines

the position of the interface (see e.g. [19]). The function Ψ(*C*) is a double

well potential with the minimal value at +1 and

*−*1 (under the assumption that the *concentration *varies between +1 and

*−*1). For instance, Ψ(*C*) =

1

4

(*C*

2

*− *1)

2

.

The equilibrium interface proﬁles can be obtained, based on the derivative of the free energy functional *F *with respect to the *concentration C *(cf,

*∂F*

*∂C*

=

Ω

(*β*Ψ

*′*

(*C*)

*− α*∆*C*)*d*Ω where the term *η ≡ β*Ψ

*′*

(*C*)

*−*

*α*∆*C *is referred to as the *chemical potential*. By minimizing the integral of the *chemical potential *within the whole domain we obtain the equilibrium proﬁles of the interface. In other words, the equilibrium proﬁles are the solutions of *η *= *β*Ψ

*′*

(*C*)

*− α*∆*C *= *const*. In one dimension, for example, the non-unique solution is *C*

1*d*

(*x*) = *tanh*(

*√*

2*ϵ*

) where the coeﬃcient *ϵ *=

*α*

*β*

is referred to as the equilibrium interface thickness and the equilibrium surface

11

tension *σ *is *σ *= *α*

∫

+

*∞*

*−∞*

(

*dC*

1*d*

*dx*

)

2

*dx *=

2

*√*

2

3

*√*

*αβ*.

The motion of the ﬂuids in a multi-phase system is, in general, due to diﬀusion and convection. Thus, the Cahn-Hilliard equation is amended to incorporate the convective process as follows

*∂C*

*∂t*

+ (**u**

*· ∇*)*C *= *∇ · *[*κ*(*C*)*∇*(*β*Ψ

*′*

(*C*)

*− α*∆*C*)]*, *in Ω *× *(0*, T *]

(3.1) where the coeﬃcient *κ*(*C*) denotes the so-called mobility, assumed to depend on the *concentration C *and **u **is the velocity. Suitable boundary conditions at the solid wall need to be added. The ﬁrst boundary condition is based on the requirement that there is no ﬂux of the *chemical potential *through solid wall surfaces. Therefore, we set Neumann boundary condition for the *che-*

*mical potential *at wall surfaces, i.e., **n**

*· ∇η *= 0, where the vector **n **denotes the unit normal vector outward the wall surface. The second boundary condition at wall surfaces corresponds to physical wetting properties (cf, e.g.,

One can solve Equation (3.1) directly, which means one needs to handle

forth order derivatives of the *concentration C*. To avoid this, by introducing the *chemical potential η *= *β*Ψ

*′*

(*C*)

*− α*∆*C *as another variable, one can rewrite the time-dependent Cahn-Hilliard equation as a coupled system of two PDEs, as follows,

*∇ · *(*κ*(*C*)*∇η*) =

*∂C*

*∂t*

*η *= *β*Ψ

*′*

+ (**u**

*· ∇*)*C,*

(*C*)

*− α*∆*C,* in Ω in Ω

*× *(0*, T *]

*× *(0*, T *]

(3.2)

(3.3) with suitable boundary and initial conditions for the *concentration C *and the *chemical potential η*.

Available results regarding the existence and uniqueness of the solution

of the C-H equations can be found in [45, 46, 25].

In Equation (3.2), the vector **u **denotes the velocity. The term **u**

*· ∇*

presents the convective eﬀect of the ﬂuid’s motion and constitutes the coupling of the Cahn-Hilliard equation and the time-dependent incompressible

Navier-Stokes (N-S) equations, which are discussed in Chapter 4.

**3.2**

**Discretization using the finite element method**

**(FEM)**

A standard technique to discretize the Cahn-Hilliard equations (3.2)-(3.3)

is to use FEM with the same ﬁnite element space for both variables, *η *and

*C*. The discrete form of the weak formulation reads as follows.

Find *η*

*h*

*, C h*

*∈ X h*

*⊂ H*

1

(Ω) satisfying

(*η*

*h*

*, v h*

)

*− β*(Ψ

*′*

(*C*

*h*

)*, v*

*h*

)

*− α*(*∇C*

*h*

*,*

*∇v h*

) = 0*,*

*κ*(

*∇η h*

*,*

*∇v h*

) + (

*dC*

*d*t

*h*

*, v h*

) + ((**u**

*h*

*· ∇*)*C*

*h*

*, v h*

) = 0*.*

(3.4)

12

for any test function *v*

*h*

*∈ X h*

. Let

*{ϕ i*

*}*

1

*≤i≤N*

the discrete solutions are of the form *C*

*h*

=

∑

*N*

*i*=1

*C i ϕ i*

and *η*

*h*

=

∑

*N h*

*i*=1

, then

*η i ϕ i*

The semi-discrete form of the weak formulation of the Cahn-Hilliard equa-

.

tion (3.4) can be presented in matrix form as follows:

Find the solution vectors **C**(*t*) =

*{C i*

(*t*)

*}*

*N*

*i*=1 and *η*(*t*) =

*{η i*

(*t*)

*}*

*N*

*i*=1 to satisfy

*M η*(*t*)

*− βf*(**C**(*t*)) *− αK***C**(*t*) = 0*,*

*κKη*(*t*) + *M*

*d***C**(*t*)

*dt*

+ *W ***C**(*t*) = 0

(3.5) where the vector *f *(**C**(*t*)) =

*{f i f i*

(**C**(*t*)) = (Ψ

*′*

(**C**(*t*))*, ϕ*

*i*

(**C**(*t*))

*}*

*N*

*i*=1 and its entries are deﬁned as

). In Equation (3.5), the involved matrices are *M *,

*K *and *W *, which are correspondingly the symmetric and positive deﬁnite mass matrix, the symmetric and semi-positive deﬁnite stiﬀness matrix and the matrix, arising from the discretization of the convection term, which is nonsymmetric, and are denoted by

*M*

=

*{M ij*

*}*

*N*

*i,j*=1

*K*

=

*{K ij*

*}*

*N*

*i,j*=1

*W*

=

*{W ij*

*}*

*N*

*i,j*=1

=

*{*(*ϕ*

*i*

*, ϕ j*

)

*}*

*N*

*i,j*=1

*,*

=

*{*(*∇ϕ*

*i*

*,*

*∇ϕ j*

)

*}*

*N*

*i,j*=1

*,*

=

*{*(**u**

*h*

*· ∇ϕ i*

*, ϕ j*

)

*}*

*N*

*i,j*=1

*.*

A method to discretize in time, often used in many numerical simulations, is the *θ*-method, *θ*

*∈ *[0*, *1] (cf, e.g., [2]). Consider a sequence

let **X**

(*k*)

=

[

*η*

*{t k*

(*k*)

0

= 0, *t*

*k*+1

= *t*

*k*

+ ∆*t*

*k*

, and denote the vector of total unknowns. The fully dis-

**C**

(*k*) cretized Cahn-Hilliard equations, which need to be solved at the *k*th timestep (*k *= 0*, *1*, ...*) is

Find **X**

(*k*)

*∈ *R

2*N* satisfying

*F*

[

(*k*)

(**X**

(*k*)

)

*≡*

*θ*∆*t*

*k*

*κKη*

(*k*)

+*M ***C**

(*k*)

+*θ*∆*t*

*k*

*M η*

(*k*)

*W ***C**

(*k*)

*−βf*(**C** (*k*)

)

*−αK***C** (*k*)

+(1

*−θ*)(∆*t*

*k*

*κKη*

(*k*

*−*1)

+∆*t*

*k*

*W ***C**

(*k*

*−*1)

)

*−M***C**

(*k*

*−*1)

]

= **0***.*

(3.6)

For instance, for *θ *= 1 the scheme corresponds to the backward Euler method (which is ﬁrst order accurate in time) and for *θ *= 1*/*2 the scheme corresponds to the Crank-Nicolson method (with second order accuracy in time).

Due to the presence of the nonlinear term *f *(**C**

(*k*)

), some linearization technique has to be used. For this purpose, Newton’s method is most often used. It is implemented via an iterative procedure as follows. At the

*k*th time-step, we start with an initial guess **X**

(*k,*0)

. An update ∆**X**

(*k,s*) of

**X**

(*k,s*) at *s*th Newton step (*s *= 0*, *1*, ...*) is computed by solving the system

13

*F*

(*k*)

*′*

(**X**

(*k,s*)

)∆**X**

(*k,s*)

=

*−F*

(*k*)

(**X**

(*k,s*)

), where **X**

(*k,s*) is the approximate solution on the *s*th Newton step and the approximate solution on the next

Newton step is formed as **X**

(*k,s*+1)

= **X**

(*k,s*)

+ ∆**X**

(*k,s*)

. The above iteration is repeated until the stopping criterion is met.

Here *F*

(*k*)

*′*

(**X**

(*k,s*)

) is the Jacobian matrix of *F*

(*k*)

(**X**

(*k,s*)

), which is of the form

*F*

(*k*)

*′*

(**X**

(*k,s*)

) =

[

*θM*

*θ*∆*t*

*k*

*κK*

*−θβJ*(**C** (*k,s*)

)

*− θαK*

*M *+ *θ*∆*t*

*k*

*W*

]

*,*

(3.7) where *J *(**C**

(*k,s*)

) is the Jacobian of the nonlinear term *f *(**C**

(*k,s*) chosen function Ψ

*′*

(*C*) = *C*

3

). For the

*− C*, the explicit expression of the matrix

*J *(**C**

(*k,s*)

In summary, at the *k*th time-step, the Newton iterations comprise a sequence of approximate solutions by solving the linear systems with the nonsymmetric Jacobian matrices *F*

(*k*)

*′*

(**X**), which are of two-by-two block form. We note, that, due to the block

*−θβJ − θαK *in (3.7), the matrix

*F*

(*k*)

*′*

(**X**) is indeﬁnite and even may become singular. Therefore, in general, inexact Newton’s method (with an approximate Jacobian matrix) should be used. This issue, however, falls out of the scope of the present work.

**3.3**

**Preconditioning technique**

As an illustration of an eﬃcient preconditioner for the C-H equations, we

summarize here the results from [13], see also the references therein. With-

out loss of generality we choose *θ *= 1 (the implicit Euler Scheme) and the

Jacobin matrix in (3.7) reads as follows:

*A*

*CH*

=

[

*A*

11

*A*

12

*A*

21

*A*

22

]

=

[

*M*

*−J − ϵ*

2

*K*

*δK*

*M *+ ∆*t*

*k*

*W*

]

*,*

(3.8) where the coeﬃcient *δ *= ∆*t*

*k*

*κ *and *ϵ*

2

*β *= 1).

The matrix

*A*

*CH*

= *α/β *(for simplicity we choose

is ﬁrst simpliﬁed. It has been discussed in [13] that

for small enough ∆*t*

*k*

relative to *h*, the inﬂuence of the blocks *J *and *W* diminishes and the idea to neglect those arises. The simpliﬁed matrix is of the form

*A*

*CH*0

=

[

*M*

*−ϵ*

2

*K*

]

*δK M*

*.*

(3.9)

Further, it turns out that the matrix

*A*

*CH*0

=

[

*M*

*−ϵ*

2

*K*

*δK*

*M *+ 2*ϵ*

*δK*

]

(3.10) is an optimal preconditioner for

*A*

*CH*0

*−*1

*CH*0

*A*

*CH*0 and all the eigenvalues of the prebelong to the interval [0.5, 1]. This idea

14

originates from [3] when solving symmetric complex systems, rewritten as

twice larger real systems. We point out that even though the system matrix

(3.9) is of two-by-two block form, the standard block-factorization methods

are not that eﬃcient since they require an approximation of the Schur complement, which is not an easy task. Here, there is no need to approximate

the Schur complement. In [13] the systems with the matrix (3.8) are solved

*A*

*CH*0

Theoretical analysis and numerical results about the preconditioner b

*CH*0 can be found there.

15

16

**Chapter 4**

**The incompressible**

**Navier-Stokes equations**

We now turn to the solution of the incompressible Navier-Stokes (N-S) equations using preconditioned iterative solution methods which, as already mentioned, is the main focus of this thesis. Since the solution of the N-S equations is considered as a part and in the context of the coupling with the

Cahn-Hilliard equations, in general, the incompressible N-S equations have to be considered in their full complexity, including time-dependence, variable viscosity and variable density. The formulation reads as follows.

*ρ*(

*∂***u**

*∂*t

+ **u**

*· ∇***u**) *− ∇ · *(2*µ***Du**) + *∇p *= **f ***− η∇C, *in Ω *× *(0*, T *]

(4.1)

*∂ρ*

+

*∇ · *(*ρ***u**) = 0*, *in Ω *× *(0*, T *]

(4.2)

*∂*t

*∇ · ***u **= 0*, *in Ω *× *(0*, T *]

(4.3) with some given boundary and initial conditions for **u**. The operator **Du **=

(

*∇***u **+ *∇*

*T*

**u**)*/*2 denotes the rate-of-strain tensor for Newtonian ﬂuids. The coeﬃcient *µ *denotes the dynamic viscosity and *ρ *denotes the density. The term *η*

*∇C *denotes the surface tension force presented in its potential form

(see e.g., [39]) and constitutes the coupling with the Cahn-Hilliard equations

(3.2)-(3.3). Equation (4.1) represents the conservation of momentum and

Equation (4.2) represents the conservation of mass. Equation (4.3) is the

incompressibility condition.

The reason we need to treat density and viscosity as variable is that even though these remain constant within each phase, however they vary in

the interfacial region, which evolves with time and in space (cf, e.g., [23]).

Therefore, density and viscosity can be seen as smooth functions of the space position and time in the whole computational domain.

In this study, we approach the solution of the N-S equations, starting from the stationary Navier-Stokes problem with constant viscosity and

17

density, where the expertise in constructing preconditioners to the corresponding discrete equations is most complete and developed, and gradually increase the complexity by next considering the variable viscosity and ﬁnally address variable density. Another reason to consider the stationary incompressible Navier-Stokes problem with constant and variable viscosity in more detail is that due to the presence of a mass matrix, the linear system arising from the time-dependent Navier-Stokes problem is relatively better conditioned than that arising from the stationary Navier-Stokes problem.

**4.1**

**Stationary incompressible N-S equations with constant viscosity**

We start by considering the stationary incompressible N-S equations with constant viscosity, which read as following:

*−ν*∆**u **+ (**u ***· ∇*)**u **+ *∇p *= **f***, *in Ω

*∇ · ***u **= 0*, *in Ω

(4.4) with some appropriate boundary conditions. Here the coeﬃcient *ν *denotes the so-called kinematic viscosity deﬁned as *ν *= *µ/ρ *and we assume that the density is constant (*ρ *= 1 for simplicity). Due to the presence of the nonlinear convection term, i.e., **u**

*· ∇***u**, some linearization technique need to

be used, e.g., the Newton or Picard method (see [26]). Here we choose the

Picard method, which consists of a sequence of approximate solutions of the linear Oseen’s problem with constant viscosity, and reads as follows:

At each Picard iteration, ﬁnd **u **: Ω

*× *(0*, T *] *→ *R

*d*

and *p *: Ω

*× *(0*, T *] *→ *R satisfying

*−ν*∆**u **+ (**w ***· ∇*)**u **+ *∇p *= **f***, *in Ω

*∇ · ***u **= 0*, *in Ω

(4.5) subject to suitable boundary conditions for **u **on *∂*Ω. Here **w **= **u**

(*k*

*−*1) is the velocity, which has been computed in the previous Picard iteration, and is updated at every Picard iteration.

Let **H**

1

*E*

=

*{***u ***∈ H*

**0 **on *∂*Ω

*D*

*} *and **X**

*h*

*E*

1

(Ω)

*d*

*| ***u **= **w **on *∂*Ω and *P*

*h*

*D*

*} *and **H**

1

*E*

0

=

*{***v ***∈ H*

1

(Ω)

*d*

*| ***v **= are the ﬁnite dimensional subspaces of **H**

1

*E*

and *L*

2

(Ω). The discrete form of the weak formulation of (4.5), deﬁned for

ﬁnite-dimensional spaces, reads (cf, e.g., [26]):

Find **u**

*h*

*∈ ***X**

*h*

*E*

*⊂ ***H**

1

*E*

and **p**

*h*

*∈ P h*

*⊂ L*

2

(Ω) such that

*ν*(

*∇***u**

*h*

*,*

*∇***v**

*h*

) + ((**w**

*· ∇*)**u**

*h*

*, ***v**

*h*

)

*− *(*∇ · ***v**

*h*

*, ***p**

*h*

) = (**f**

*h*

*, ***v**

*h*

)*,*

(

*∇ · ***u**

*h*

*, ***q**

*h*

) = 0*.*

(4.6)

18

for all **v** of **X**

*h*

*E h*

*∈ ***X** and

*{ϕ h*

*E*

0

*i*

*}*

*⊂ ***H**

1

*≤i≤n p*

1

*E*

0 and all **q**

*h*

*∈ P h*

. Let be the nodal basis of *P*

*{φ h i*

*}*

1

*≤i≤n u*

such that, be the nodal basis

**u**

*h*

=

*i*=1

**u**

*i*

*⃗ i*

*, ***p**

*h*

=

*i*=1

*p i ϕ i*

*,*

where *n*

*u*

and *n*

*p*

are the total number of unknowns for the velocity and pressure. The linear systems, arising from the discrete weak formulation

(4.6) with LBB stable FEM (cf, e.g., [26]), are of the form

[

*A B*

*T*

*B O*

] [

**u**

*h*

]

**p**

*h*

=

[

**f**

]

**g**

or

*A*

*CV*

**x **= **b***,* (4.7) where the pivot matrix *A*

*∈ *R

*n u*

*×n u*

corresponds to the discrete convectiondiﬀusion operator, i.e., *A *= *νL *+ *N *. The matrix *L *denotes the discrete

Laplacian matrix, which is symmetric and positive deﬁnite (after Dirichlet boundary conditions applied), and the matrix *N *denotes the discrete convection matrix, which is nonsymmetric. The matrix *B*

*∈ *R

*n p*

*×n u*

corresponds to the discrete (negative) divergence operator and the matrix *B*

*T*

corresponds to the discrete gradient operator. For more details on the properties of the

above matrices one can see [26].

The system matrix

*A*

*CV*

is nonsymmetric of saddle point form. Preconditioned iterative solution methods for saddle point problems have been

studied intensively during the last 30 years (see e.g. [8, 26] and references

therein). Eﬃcient preconditioners are usually constructed based on some approximation of the factorization of the original matrix. In general, the exact factorization of a matrix of two-by-two block form is

*A *=

[

*A*

11

*A*

12

*A*

21

*A*

22

]

=

[

*A*

11

0

] [

*I*

1

*A*

*−*1

11

*A*

12

]

*A*

21

*S*

0

*I*

2

*,*

(4.8) where *I*

*A*

11

1 and *I*

2 are identity matrices of proper dimensions. The pivot block is assumed to be nonsingular and *S *= *A*

22

*− A*

21

*A*

*−*1

11

*A*

12

*T*

is the exact

Schur complement matrix. In our case, *A*

11

= *A*, *A*

12

= *B* and *A*

22

= *O*. So, *S*

*A*

*CV*

*≡ S *= *−BA*

*−*1

*B*

*T*

, *A*

21

= *B*

. The preconditioners for such matrices of two-by-two block form are either of full block-factorized form,

such as (4.9) or of block lower- or upper-triangular form, as in (4.10).

*M*

*F*

=

[ e

11

*O*

] [

*I*

1

*A*

21

*O*

*−*1

11

*I*

2

*A*

12

]

*,*

(4.9)

*M*

*L*

=

[ e

11

*O*

]

*,*

*A*

21

*M*

*U*

=

[ e

11

*A*

12

]

*.*

0

(4.10)

19

*−*1

11 denotes some approximation of *A*

*−*1

11

, given on explicit form or implicitly deﬁned via inner iterative solution methods. The matrix

*S *is some approximation of the exact Schur complement *S*.

When solving systems with the preconditioner

*M*

*F*

*−*1

11 twice. This is, clearly, computationally heavier task, compared to

*M*

*L*

and

*M*

*U*

*M*

*L*

and

*M*

*U*

*−*1

11

, we need the action is needed once.

It is shown in

are equally eﬃcient. In [7] it is pointed out that

for indeﬁnite systems, the block-triangular preconditioner,

*M*

*L*

more eﬃcient than the full block-factorized preconditioner

*M*

*F*

or

*M*

*U*

, is

. So, for the

Oseen’s problem with constant viscosity, the block-triangular preconditioner

*M*

*L*

or

*M*

*U*

is the one to choose.

For e

11

= *A*

11 and e

*M*

ditioned matrix

*M*

*−*1

*L*

*L*

*A *(*A *is deﬁned as 4.8) is of the form

*M*

*−*1

*L*

*A *=

[

*−S*

*A*

*−*1

*−*1

11

*A*

21

*A*

*−*1

11

*S*

*O*

*−*1

] [

*A*

11

*A*

21

*A*

*A*

12

22

]

=

[

*I*

1

0

*A*

*−*1

11

*I*

2

*A*

12

] where the matrices *I*

1 and *I*

2 are the identity matrices with proper dimensions, and the matrices *A*

11

and *S *are nonsingular. The results in [7] show

that (i) in this case the minimal polynomial of

*M*

*P *(

*·*) of the smallest degree for *P *(*M*

*−*1

*L*

*−*1

*L*

*A*, i.e., the polynomial

*A*) = 0 takes the form *P *= (1 *− t*)

2 and there will be at most two iterations when solving systems with the matrix

*A *using iterative solution methods with the preconditioner *M*

*L*

*−*1

11

*≈ A*

*−*1

11 and e

; (ii) in

*≈ S*, the eigenvalues of *M*

*−*1

*L*

*A*

are located in disks and the radii of the disks is controlled by making a suﬃcient number of inner iterations when solving systems with the pivot block matrix *A*

11 of *S*. Thus, we can see that the quality of the preconditioner

*M*

*L*

of the matrix

*A *depends on the accurate solutions of the pivot block matrix and how well the Schur complement matrix is approximated. Compared with the accurate solutions of *A*

11

, the most challenging task, however, is how to construct numerically and computationally eﬃcient approximations of the

Schur complement matrix, which is in general dense and it is not practical to form it explicitly.

The research on Schur complement approximations for the Stokes and

Oseen’s problem with constant viscosity is quite active during the past

decades (see [49] and [26] and references therein). In Paper II we do a

short survey of the known approximations of the Schur complement. The included (problem-dependent) Schur complement approximations may be

costly to apply, e.g., the BFBt approximation [27] or may need the con-

struction of an artiﬁcial convection-diﬀusion operator on the ﬁnite element space for the pressure, e.g., the pressure convection-diﬀusion approximation

[40]. These approximations are fairly robust with respect to the discretiza-

tion and problem parameters, i.e., the mesh size *h *and the viscosity *ν*. In

20

Paper I we contribute to the search for eﬃcient Schur complement approximations by trying and analyzing the element-by-element Schur complement approximation. This preconditioner is implemented based on the local features of the ﬁnite element discretization, and is of the form

*S*

*EBE*

=

*k*=1

*R*

*T k*

*S e*

*R k*

(4.11) where *R*

*k*

are the Boolean matrices which prescribe the local-to-global correspondence of the degrees of freedom and *S*

*e*

is the local Schur complement on each macro element. The total number of macro elements is denoted by *n*

*E*

.

From the formula (4.11) we see that the construction of this approximation

is relatively cheap. For a uniform mesh, we only need to compute the local

Schur complement on one macro element and assemble it for all the macro el-

ements. In several works (see e.g. [41, 6]) it has been shown that, in the case

of symmetric and positive deﬁnite matrices, split into a two-by-two block form, based on consecutive regular mesh reﬁnements, *S*

*EBE*

is a high quality approximation of the exact Schur complement *S*, i.e., (1

*−ζ*

2

)*S*

*≤ S*

*EBE*

*≤ S*, where *ζ *is a positive constant, strictly less that 1, independent on the mesh size and easily computable. (Here the notation *K*

*≤ Z *means that the matrix *Z*

*− K *is positive semi-deﬁnite.) In Paper I this method is used to approximate the Schur complements of the system matrices arising from the Cahn-Hilliard and Oseen’s problem, where the system matrices are nonsymmetric and the two-by-two form of the element matrices is due not to the mesh reﬁnement but to the fact that we deal with a system of two equations. Although this approximation is not robust with respect to the mesh size and the viscosity, it is still attractive in some cases because of its low computational cost.

Aiming at ﬁnding some preconditioner which is fully independent of the mesh size and the viscosity, we turn to the so-called augmented Lagrangian

(AL) approach (see e.g. [29]). Following this approach we ﬁrst transform

the linear system (4.7) into an equivalent one with the same solution, which

is of the form

[

*A *+ *γB*

*T*

*W*

*B*

*−*1

*B B*

*T*

0

] [

**u**

*h*

]

**p**

*h*

=

**g**

] or

*A*

*CV*

(4.12) where *γ > *0 and *W *are suitable scalar and matrix parameters. The modiﬁed

*T*

*W*

*−*1

*B ***g**. It is clear that the transformation

(4.12) holds for any value of *γ*, including *γ *= 1 or *γ*

*≪ *1, and any nonsingular

matrix *W *. In paper [9] and [10] the AL type preconditioners are proposed for

the transformed system (4.12), which are of block lower- or upper-triangular

form

*M*

*Lcv*

=

[

*A*+*γB*

*T*

*W*

*B*

*−*1

*B*

0

*−*

1

*γ*

*W*

] or

*M*

*U cv*

=

[

*A*+*γB*

*T*

0

*W*

*−*1

*B B*

*T*

*−*

1

*γ*

*W*

]

(4.13)

21

Now we can explain the purpose of doing the transformation (4.12).

*A*

*CV*

(4.12) with its AL type preconditioner (4.13)

and the general two-by-two block matrix

*A *(4.8) with its block triangular preconditioner (4.10), we can see that the Schur complement *S*

*A*

*CV*

*−B*(*A*+*γB* mated by

*−*

*T*

1

*γ*

*W*

*−*1

*B*)

*−*1

*B*

*T*

*A*

*CV*

*W *, where the matrix *W *can be the pressure mass matrix as

shown in [9] or even be the identity matrix as shown in [5].

=

*A*

*CV*

*T*

*W*

*−*1

*B*, becomes increasingly ill-conditioned with

*γ*

*→ ∞*, which contradicts to the requirement that *γ *needs to be large.

Numerical experiments in paper II and [9] illustrate that for moderate va-

lues of *γ*, such as *γ *= 1, the AL type preconditioners with *W *= *M*

*p*

(the pressure mass matrix) work eﬃciently and do not depend on the mesh size and the viscosity. This observation is very important because it eases the implementation of the AL type preconditioners.

On the other side, although we circumvent the diﬃculties to approxi-

mate the Schur complement of the original system (4.7), we move the cha-

llenge to how to eﬃciently solve systems with the modiﬁed pivot matrix

*A *= *A *+ *γB*

*T*

*W*

*−*1

*B*, which is much denser than *A *and is also a Schur complement matrix.

Direct methods are used in many research papers, which means that the pivot block has to be explicitly computed ﬁrst, and then can be factorized. This becomes increasing unacceptable with the size of problems turning bigger and bigger. In other studies, for instance, in

[9] and [10], a problem-dependent multigrid method has been derived. In

*T*

*W*

*−*1

*B *are derived, which work eﬃciently, however, requiring that the values of *γ *have to be relatively small, empirically determined as 0*.*01 even 0*.*001. The latter contradicts to the requirement that *γ *needs to be large so that the AL type preconditioners

(4.13) work well for the transformed system (4.12). Our main contribution

regarding the AL type preconditioners is that in Paper II we explain the above behavior through a more general framework. The analysis reveals that if we attempt to balance the inner solution (preconditioners to the

*A*

*CV*

for the Schur complement, i.e.,

*−BA*

*A*

*CV*

), then ﬁnding a good preconditioner

*−*1

*B*

*T*

, of the original system matrix

22

**4.2**

**Stationary incompressible N-S equations with variable viscosity**

Consider next the stationary incompressible Navier-Stokes equations with variable viscosity, which read as follows

*−∇ · *(2*ν*(**x**)**Du**) + (**u ***· ∇*)**u **+ *∇p *= **f***, *in Ω

*∇ · ***u **= 0*. *in Ω

(4.14) with suitable boundary conditions and the assumption that density is constant (*ρ *= 1 for simplicity). This type problems are also studied for non-

Newtonian ﬂows, where the variable viscosity may depend on the pressure

and the rate-of-strain tensor (cf, e.g., [12, 51]) or the pressure and the shear

(cf, e.g., [38, 43, 50]). In Paper III we assume that the kinematic viscosity

coeﬃcient is a smooth function, such that

0

*≤ ν*

min

*≤ ν*(**x**) *≤ ν* max

*,*

where *ν* min and *ν* max denote its minimal and maximal value. As mentioned already, the motivation of considering the variable viscosity of this form is that in the simulation of the multi-phase ﬂow the density and viscosity remain constant in each phase and vary rapidly and smoothly in the interfacial region.

There are several ways to treat the nonlinear convection term (**u**

*· ∇*)**u**

in (4.14). (i) One option is to treat it explicitly and move it into the right

hand side. This leads to a Stokes-type problem with variable viscosity and the resulting algebraic system matrix is the same on each iteration. For this formulation, block preconditioner involving a mass type matrix for the pre-

ssure is proposed and analyzed in [32]. However, the above treatment of the

nonlinear term may not be eﬃcient for convection-dominated problems. (ii)

Another option is to linearize and discretize the convection term, and incorporate the arising matrix into the system matrix. Thus, Oseen-type problem with variable viscosity arises. In Paper III, we focus on the construction of eﬃcient preconditioners for the matrices arising from Oseen-type problem with variable viscosity.

Here we also choose the Picard method to linearize (4.14) and Oseen-

type problem with variable viscosity reads as follows.

At each Picard iteration, ﬁnd **u **: Ω

*× *(0*, T *] *→ *R

*d*

and *p *: Ω

*× *(0*, T *] *→ *R satisfying

*−∇ · *(2*ν*(**x**)**Du**) + (**w ***· ∇*)**u **+ *∇p *= **f***, *in Ω

*∇ · ***u **= 0*, *in Ω

(4.15) subject to suitable boundary conditions for **u **on *∂*Ω. Here **w **= **u**

(*k*

*−*1) is the velocity which has been computed in the previous Picard iteration, and is updated at every Picard iteration.

23

The discrete form of the weak formulation of (4.15) deﬁned using ﬁnite-

dimensional spaces reads:

Find **u**

*h*

*∈ ***X**

*h*

*E*

*⊂ ***H** 1

*E*

and **p**

*h*

*∈ P h*

*⊂ L*

2

(Ω) such that

(2*ν*(**x**)**Du**

*h*

*, ***Dv**

*h*

) + ((**w**

*· ∇*)**u**

*h*

*, ***v**

*h*

)

*− *(*∇ · ***v**

*h*

*, ***p**

*h*

) = (**f**

*h*

*, ***v**

*h*

)*,*

(

*∇ · ***u**

*h*

*, ***q**

*h*

) = 0*.*

(4.16) for all **v**

**H**

1

*E*

0

*h*

*∈ ***X**

*h*

*E*

0 and **H**

1

*E*

*⊂ ***H** the nodal basis of *P*

*h*

1

*E*

0

Here we again let

*{φ i*

*}*

and all **q**

1

*≤i≤n u*

such that **u**

*h h*

=

*∈ P h n u*

. The deﬁnitions of the spaces of can be found in Chapter 4.1.

*i*=1

**u**

*i*

*φ i*

and **p**

*h h*

*E*

= and

*{ϕ n p i*

*i*=1

*} p i*

1

*≤i≤n p ϕ i*

be

, where

*n u*

and *n*

*p*

are the total number of unknowns for the velocity and pressure.

The linear systems arising from the discrete weak formulation (4.16) with

LBB stable FEM (cf, e.g., [26]) are of the form

[

*F B*

*B O*

*T*

] [

**u p**

where the system matrix

*A*

*V V h h*

]

=

=

[

[

*F*

*B*

**f g**

]

*B*

*O*

*T*

or

]

*A*

*V V*

**x **= **b***,* (4.17) is again nonsymmetric of saddle point form. The unknown vector **u**

*h*

is the discrete velocity vector and **p**

*h*

is the discrete pressure vector. Combining them together we have **x**

*T*

=

[**u**

*T h*

**p**

*T h*

]. The deﬁnitions of the matrices *B *and *B*

*T*

have been given in

Chapter 4.1. Clearly, when considering variable viscosity, the only diﬀerence in the arising matrices with FEM, compared to the Oseen’s problem with constant viscosity, is observed in the pivot block matrix *F*

*∈ *R

*n u*

*×n u*

, which, in the case of variable viscosity has the form *F *= *A*

*ν*

+ *N *. The discrete convection matric *N *has been deﬁned in Chapter 4.1 and the matrix *A*

*ν*

, which arises from the discretization of the term (2*ν*(**x**)**Du***, ***Dv**), is of the form [*A*

*ν*

]

*i,j*

= (2*ν*(**x**)**D***⃗*

*i*

*φ j*

). The matrix *A*

*ν*

is symmetric and positive deﬁnite (see more details in Paper III).

In order to construct eﬃcient preconditioners for the system matrix

*A*

*V V*

in (4.17), we consider once again the augmented Lagrangian approach.

Namely, we transform the system (4.17) into an equivalent one, which is

[

*F *+ *γB*

*T*

*B*

*W*

*−*1

*B B*

0

*T*

] [

**u p**

*h h*

]

=

**g**

] or

*A*

*V V*

(4.18)

and then propose the AL type preconditioners (4.19) to the transformed

*A*

*V V*

[

*M*

*Lvv*

=

.

*F *+*γB*

*T*

*B*

*W*

*−*1

*B*

0

*−*

1

*γ*

*W*

] or

*M*

*U vv*

=

[

*F *+*γB*

*T*

0

*W*

*−*1

*B B*

*T*

*−*

1

*γ*

*W*

]

*. *(4.19)

Our contributions to constructing the eﬃcient preconditioners of Oseentype problems with variable viscosity are that (i) in Paper III we theoretically prove that the bounds of the eigenvalues of the preconditioned matrix,

24

*−*1

*Lvv*

*A*

*V V*

with *W *= *M*

*p*

(*M*

*p*

is the pressure mass matrix) are independent of the mesh size, but dependent of the values of *ν* min and *ν* max

. The derived bounds are generalizations of those, derived for constant viscosity,

as shown in [27] and Paper II. (ii) Numerical experiments in Paper III illus-

*M*

*Lvv*

with *W *= *M*

*p*

*A*

*V V*

and *γ *= 1 still for a large range of the values of *ν* min and *ν* max

.

However, how to eﬃciently solve systems with the modiﬁed pivot block

*F *= *F *+ *γB*

*T*

*W*

*−*1

*A*

*V V*

most diﬃcult part in the application of the AL type preconditioner. This open question is discussed in Chapter 5.

**4.3**

**Incompressible N-S equations with variable density**

One often used technique to solve the time-dependent incompressible Navier

Stokes equations with variable density is via some operator splitting method.

Most of the known splitting methods decouple the diﬀusion operator and the incompressibility constrain. These are originally developed from the

*simple *projection method, which is originally proposed by Chorin [22] and

Temam [55] and also referred to as the *non-incremental pressure-correction*

algorithm. To understand the splitting technique, we describe the original simple projection algorithm in some detail and for simplicity let the timestep ∆*t *be constant, i.e., ∆*t *= *T /N *. Then, the points in time are *t*

*k*

= *k*∆*t* for *k *= 0*, *1*, ..., N *. Neglecting the nonlinear convection term in the N-S equations for simplicity and using the implicit Euler time stepping scheme, the non-incremental pressure-correction algorithm applied to solve the timedependent Stokes problem with constant density reads as follows:

Set **u**

0

= **u**

0

, then for all time-steps *t*

*k*+1

*, k*

*≥ *0, compute (˜**u**

*k*+1

, **u**

*k*+1

, *p*

*k*+1

) by solving

*ρ*

∆*t*

**u**

*k*+1 *− ***u**

*k*

)

*− µ*∆˜**u** *k*+1

= **f**(*t*

*k*+1

)*,*

**u**

*k*+1

= 0 on *∂*Ω*,*

(4.20) and then

1

∆*t*

(**u**

*k*+1

*− *˜**u**

*k*+1

) +

1

*ρ*

*∇p*

*k*+1

= 0*,*

**u**

*∇ · ***u** *k*+1

*k*+1

= 0 in Ω*,*

*· ***n **= 0 on *∂*Ω*,*

(4.21) where **n **denotes the unit outward normal vector. Clearly, at each time-step this algorithm consist of two sub-steps. In the ﬁrst sub-step, one computes

25

**u**

*k*+1

*p*

*k*+1

**u**

*k*+1 is obtained, the pressure is computed by solving the following Poisson problem:

∆*p*

*k*+1

=

*ρ*

∆*t*

*∇ · *˜**u** *k*+1

*,*

*∂p*

*k*+1

*∂n*

= 0 on *∂*Ω*.*

Indeed, the second sub-step can be represented as follows:

**u**

*k*+1 and *p*

*k*+1 to compute **u**

*k*+1 by

**u**

*k*+1

+

∆*t*

*∇p*

*k*+1

*ρ*

**u**

*k*+1

**u**

*∇ · ***u** *k*+1

*k*+1

= 0 in Ω*,*

*· ***n **= 0 on *∂*Ω*.*

*,*

(4.22)

(4.23)

By introducing the Hilbert space

**H **=

*{***u ***∈ L*

2

(Ω)

*| ∇ · ***u **= 0 in Ω; **u ***· ***n **= 0 on *∂*Ω*},* the second sub-step can be equivalently rewritten as the projection step

**u**

*k*+1

= *P*

**H**

˜

*k*+1

, where *P*

**H**

is the *L*

2

-projection onto **H**. This is the reason to refer this method and its other variants as projection methods.

The above algorithm is of ﬁrst order accuracy in time for the velocity and pressure. The above operator splitting scheme introduces a splitting error

of order *O*(∆*t*) (cf, e.g., [34]), and therefore, there is no improvement of the

overall accuracy if one uses higher order time discretization schemes. There are many variants derived from the non-incremental pressure-correction algorithm, e.g., the standard incremental pressure-correction algorithm, the rotational incremental pressure-correction algorithm and so on. Higher order scheme, e.g., the backward diﬀerence formula of second order (BDF2) can be used to approximate the time derivative in the standard/rotational incremental pressure-correction algorithms. which are of second order accuracy in time for the velocity. For a comprehensive view of the projection

methods for the incompressible ﬂows, we refer to [34].

For the variable density, most of the already known splitting algorithms are similar to the non-incremental pressure-correction algorithm introduced above. They also consist of two sub-steps at each time-step. In the ﬁrst

step, one need to solve problems similar to (4.20), where the pressure is

treated explicitly or ignored. In the second step one need to solve problems

follows:

*−∇ · *(

1

*ρ*

*k*+1

*∇*Φ) = Ψ*,*

*∂*Φ

*k*+1

*∂n*

= 0 on *∂*Ω*,* (4.24) where *ρ*

*k*+1 is the approximation of the variable density at discrete time *t*

*k*+1 and Ψ is some right hand side with varies at every time-step. The variable

26

Φ may be the pressure or some other related scalar.

Stable algorithms proposed for solving the incompressible N-S equations with variable density

Due to the variable density one need to assemble and solve a system

with the variable coeﬃcient stiﬀness matrix like (4.24) in the second sub-

step. Clearly, it is time consuming. In [36] eﬃcient algorithms are proposed

for solving the incompressible ﬂows with variable density, which involve the assemble of the stiﬀness matrix only once and solving the Poisson problem

instead of the Poisson-type problem like (4.24) in the second sub-step. In

future study, one research area is the construction of eﬃcient preconditioners

of the systems arising from the problems similar to (4.20) in the ﬁrst sub-step

of the algorithms introduced in [36].

Splitting the incompressibility constrain and the diﬀusion operator re-

duces the computational complexity to solving problems as in (4.24) and

as in (4.20). However, the splitting error and accuracy deﬁciency are the

main limitations of the known splitting techniques. The existing algorithms, which have been proved to be stable, are of up to second order in time for the velocity, which means that we have to use relatively small time-steps to keep the accuracy of the solutions. Another research area in the future is ﬁnding some eﬃcient solution methods, which have higher accuracy for the solutions in time and one possible method is very brieﬂy introduced in

Chapter 7.

27

28

**Chapter 5**

**Computational challenges**

In Chapters 4.1 and 4.2 we consider iterative solution methods for the

Navier-Stokes problems with constant and variable viscosity. We choose the AL type preconditioner and there are two computational issues which need to be considered.

(1) When constructing and solving systems with the AL type preconditioners, we need an eﬃcient and cheap approximate inverse of the pressure mass matrix.

In Paper II, based on local features of the

Finite Element discretization (see e.g. [30, 31, 58]), we derive and

analyse the so-called element-by-element sparse approximate inverse of the pressure mass matrix. Numerical experiments in Paper II and

Paper III show that this approximate inverse of the pressure mass matrix, involved in the AL type preconditioners, works well for both the constant and variable viscosity cases on a low computational cost. For

the related topics on the sparse approximate inverse we refer to [11],

*γB*

*T*

*γB*

*T*

*W*

*−*1

*W*

*−*1

*B *(in the variable viscosity case) in (4.19) is still an open

question. We propose an algorithm to compute the exact or approxi-

structed based on the block ISM algorithm and numerical experiments there show that the block ISM algorithm is eﬃcient both in terms of the number of preconditioned iterations and the constructing time.

one possible approach to resolve this challenging open question. In an unpublished work, we have implemented and tested the performance of the ISM method in parallel, using a block-version of it and the standard BLAS operations.

29

When solving the coupled system comprising the incompressible Navier-

Stokes (N-S) and Cahn-Hilliard (C-H) equations, there are two challenging issues to be considered.

(3) In order to decrease the number of the degrees of freedom, the adaptively reﬁned and de-reﬁned meshes are needed. How to determine the domains where the mesh needed to be reﬁned or de-reﬁned and how to choose the stopping tolerances for the reﬁnement and de-reﬁnement

are the main challenges. These issues are discussed in [59] and [24].

(5) The smaller time-steps result in better conditioned system matrices arising from the time-dependent N-S and C-H equations and it is easier to solve systems with these matrices. However, smaller time-steps require more computation, especially when the simulation time interval is large. Thus, ﬁnding suitable time-steps balancing the aspects mentioned above is also quite important and challenging.

30

**Chapter 6**

**Summary of Papers**

**6.1**

**Paper I**

In this paper we consider preconditioned iterative solution methods for the numerical simulation of multi-phase ﬂow, which is governed by the time-dependent Cahn-Hilliard (C-H) and incompressible Navier-Stokes (N-

S) equations. The test problems considered in this paper consist of the linearized stationary N-S problem, i.e., Oseen’s problem with constant viscosity in 2D and a moving interface with constant speed governed by the C-H equations also in 2D. The matrices arising from the ﬁnite element discretization of the above two problems are nonsymmetric of two-by-two block form.

The Schur complements of the so-arising matrices are approximated via the element-by-element Schur complement approximation method, which is

shown in several works (e.g. [41, 6]) to be a cheap technique resulting in high

quality Schur complement approximations for the symmetric and positive deﬁnite matrices. In this paper, a framework is suggested to study the quality of the element-by-element Schur complement approximation for general nonsymmetric matrices. The numerical experiments show that due to its low computational cost, the element-by-element Schur complement approximation is still an attractive technique to precondition the exact Schur complement for the considered test problems, in particular for the N-S problems, which are not strongly convection-dominated.

**6.2**

**Paper II**

In this paper we mainly focus on constructing eﬃcient preconditioners for the nonsymmetric matrices of saddle point form arising from the ﬁnite element discretization of the Oseen’s problem with constant viscosity. Aiming at ﬁnding some eﬃcient preconditioners, which are fully robust with respect to the mesh size and the viscosity, we choose the augmented Lagrangian

(AL) method, where the original system is transformed into an equivalent

31

one and a block triangular preconditioner is used for the transformed system matrix. The AL type approach involves a scalar parameter *γ *and an approximate inverse of the pressure mass matrix. The main challenge is how to eﬃciently solve systems with the modiﬁed pivot block matrix of the transformed system matrix, which is much denser than the original one. There exist some preconditioners for the modiﬁed pivot block matrix, which work

well for relatively small values of *γ*, such as 0*.*01 and 0*.*001 (see e.g. [10]).

However, the value of *γ *needs to be large to make the AL type preconditioner work eﬃciently for the whole system matrix. The main contribution in this paper is that via a more general framework we analyse the inﬂuence of the value of *γ *on both the convergence rates of the outer (for the whole system matrix) and the inner (for the pivot block matrix) solution methods.

Another contribution is that we propose an algorithm to compute the exact

(or an approximation of the) inverse of the modiﬁed pivot matrix, based on the inverse Sherman-Morrison (ISM) formula. The multiplicative preconditioner, constructed via the block ISM algorithm, is shown to be eﬃcient in

terms of the construction time and preconditioning time (see e.g. [21]).

**6.3**

**Paper III**

In this paper we consider the construction of eﬃcient preconditioners for the matrices arising from the ﬁnite element discretization of Oseen-type problem with variable viscosity. We apply the augmented Lagrangian approach. In Paper III we prove that the AL type preconditioner involving the pressure mass matrix is independent of the mesh size, but depends on the maximal and minimal values of the viscosity, i.e., *ν* max

*ν*

max

= *ν* min and *ν* min

. For

= *ν*, the theoretical results derived in this paper coincide with those when using the augmented Lagrangian approach and the AL type pre-

conditioner for the Oseen’s problem the constant viscosity (see e.g. [28]).

Numerical experiments in Paper III show that the AL type preconditioner works eﬃciently for a large rang of the values of *ν* max and *ν* min

. However, the eﬃcient solution methods for the modiﬁed pivot block matrix of the transformed system matrix is more challenging compared to the case of constant viscosity.

32

**Chapter 7**

**Summary and future work**

A future research direction is the fast solution method for the time-dependent incompressible Navier-Stokes equations with variable density and viscosity.

There are two approaches for this problem. The ﬁrst one is to use the socalled splitting method to decouple the solution for density, velocity and the pressure. This method is also referred to as the projection method, which consists of three sub-steps. In the ﬁrst two sub-steps one compute the solution for density and velocity. In the third sub-step one need to solve a

Poisson problem to compute the solution for pressure. More details about

the splitting method can be found in [36]. Eﬃcient solution methods for the

Poisson problem have been studied intensively, and the parallel algebraic

multi-grid (AMG) (see e.g. [56]) is an eﬃcient method to solve the Poisson

problem. We consider iterative solution methods to solve the systems with the matrices arising from the ﬁrst two sub-steps. Eﬃcient preconditioners for these so-arising matrices need a further study. The known splitting methods, which have been proved to be stable, are up to the second order accuracy in time, therefore, relatively small time-steps are needed to keep the accuracy of the solution.

Aiming at using large time-steps to reduce the simulation time while keeping the accuracy of the solutions within certain bounds, we may consider another approach, namely, to represent the system governing the incompressible and time-dependent Navier-Stokes problem as a system of diﬀe-

rential algebraic equations (DAEs) (see e.g. [48]). In this way high-order

time integration methods and large time-steps can be used. The solution methods for DAEs and their applications to the Navier-Stokes problems need to be considered in future.

33

34

**Acknowledgements**

I am very grateful to my supervisor Maya Neytcheva for her support and encouragement, and in particular, for sharing her expertise. Many thanks manuscript and providing useful comments and suggestions. I thank Professor Owe Axelsson and Professor Serra Capizzano Stefano for their very valuable comments for the Paper II. I acknowledge discussions with Martin

Kronbichler and Petia Boyanova on the ﬁnite element method.

35

36

**Bibliography**

[1] O. Axelsson. Precondioning of indeﬁnite problems by regularization.

*SIAM J. Numer. Anal.*, 16:58-69, 1979.

[2] O. Axelsson. Finite Diﬀerence Methods, *The Encyclopedia of Com-*

*putational Mechanics, Part I Fundamentals, Ch. 2*, Wiley, 2004.

[3] O. Axelsson, A. Kucherov. Real valued ierative methods for solving complex symmetric linear systems. *Num. Lin. Alg. Appl.*, 7:197-218,

2000.

[4] O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods, II. *SIAM J. Numer. Anal.*, 27:1569-1590, 1990.

[5] O. Axelsson, R. Blaheta. Preconditioning of matrices partitioned in two-by-two block form: Eigenvalue estimates and Schwarz DD for mixed FEM. *Num. Lin. Alg. Appl.*, 17:787-810, 2010.

[6] O. Axelsson, R. Blaheta, M. Neytcheva. Preconditioning for boundary value problems using elementwise Schur complements. *SIAM J.*

*Matrix Anal. Appl.*, 31:767-789, 2009.

[7] O. Axelsson, M. Neytcheva. A general approach to analyse preconditioners for two-by-two block matrices. Department of Information

Technology, Uppsala University, Sweden. TR 2010-029, 2010.

[8] O. Axelsson, M. Neytcheva. Eigenvalue estimates for preconditioned saddle point matrices. *Num. Lin. Alg. Appl.*, 13:339-360, 2006.

[9] M. Benzi, M.A. Olshanskii. An augmented Lagrangian-based approach to the Oseen problem. *SIAM J. Sci. Comput.*, 28:2095-2113,

2006.

[10] M. Benzi, M.A. Olshanskii, Z. Wang. Modiﬁed augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations, *Int. J. Numer. Meth. Fluids.*, published online in Wiley Inter-

Science (www.interscience.wiley.com), 2010.

37

nonsymmetric linear systems. *SIAM J. Sci. Comput.*, 19:968–994,

1998.

[12] M. Bercovier, M. Engelman, A ﬁnite element method for incompressible non-Newtonian ﬂows. *J. Comput. Phys.*, 36:313-326, 1980.

[13] P. Boyanova, M. Do-Quang, M. Neytcheva. Solution methods for the Cahn-Hilliard equation discretized by conforming and nonconforming ﬁnite elements. Technical Report, Department of Information Technology, Uppsala University, Sweden, 2011.

[14] R. Bru, J. Cerdn, J. Mar´ın, J. Mas. Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula.

*SIAM J. Sci. Comput.*, 25:701–715, 2003.

tion. *SIAM J. Sci. Comput.*, 30:2302-2318, 2008.

[16] E. Chow, Y. Saad. Approximate inverse techniques for blockpartitioned matrices. *SIAM J. Sci. Comput.*, 18:1657–1675, 1997.

[17] J.W. Cahn. On spinoidal decomposition. *Acta Metallurgica*, 9:795-

801, 1961.

[18] J.W. Cahn, J. Hillard. Free energy of a nonuniform system. I. Interﬁcial free energy. *J. Chem. Phys.*, 28:258-267, 1958.

[19] A. Carlson, M. Do-Quang, G. Amberg. Droplet dynamics in a bifurcating channel. *Int. J. Multiphase Flow*, 36:397-405, 2010.

[20] A. Carlson, M. Do-Quang and G. Amberg, Modeling of dynamic wetting far from equilibrium, *Phys. Fluids*, 21, 2009.

inverse preconditioners for sparse nonsymmetric linear systems. *Elec-*

*tron. Trans. Numer. Anal.*, 37:23-40, 2010.

[22] A.J. Chorin. Numerical solution of the Navier-Stokes equations.

*Math. Comput.*, 22:745-762, 1968.

[23] M. Do-Quang, G. Amberg. The splash of a ball hitting a liquid surface: Numerical simulation of the inﬂuence of wetting. *Phys. Fluids*,

2008.

[24] M. Do-Quang, W.Villanueva, I. Singer-Loginova, G. Amberg. Parallel adaptive computation of some time-dependent materials-related microstructural problems. *Bulletin of the Polish academy of sciences*,

55, 2007.

38

[25] C.M. Elliott, Z. Songmu. On the Cahn-Hilliard equation. *Arch. Ra-*

*tional Mech. Anal.*, 96:339-357, 1986.

[26] H.C. Elman, D.J. Silvester, A.J. Wathen. *Finite Element and Fast*

*Iterative Solvers: with Application in Incompressible Fluid Dynam-*

*ics*. Oxford University Press, Oxford, UK, 2005.

[27] H.C. Elman. Preconditioning for the steady-state Navier-Stokes equations with low viscosity. *SIAM J. Sci. Comput.*, 20:1299-1316,

1999.

[28] H. Elman, D. Silvester. Fast nonsymmetric iterations and preconditioning for Navier-Stikes equations. *SIAM J. Sci. Comput.*, 17:33-46,

1996.

[29] M. Fortin, R. Glowinski. *Augmented Lagrangian Methods: Applica-*

*tion to the Numerical Solution of Boundary-Value Problems*. North-

Holland, 1983.

[30] I. Fried. Bounds on the spectral and maximum norms of the ﬁnite element stiﬀness, ﬂexibility and mass matrices. *Int. J. Solids Struc-*

*tures*, 9: 1013–1034, 1973.

[31] I. Fried, M. Coleman. Improvable bounds on the largest eigenvalue of a completely positive ﬁnite element ﬂexibility matrix. *J. Sound*

*Vibration*, 283:487-494, 2005.

[32] P.P. Grinevich and M.A. Olshanskii, An iterative method for the stokes-type problem with variable viscosity. *SIAM J. Sci. Comput.*,

31:3959-3978, 2009.

[33] M. Grote, T. Huckle. Parallel preconditioning with sparse approximate inverses. *SIAM J. Sci. Comput.*, 18:838–853, 1997.

[34] J.L. Guermond, P. Minev, J. Shen. An overview of projection methods for incompressible ﬂows. *Comput. Methods Appl. Mech. Eng.*,

195:6011-6045, 2006.

[35] J.L. Guermond, L. Quartapelle. A projection FEM for variable density incompressible ﬂows. *J. Comput. Phys.*, 165:167-188, 2000.

[36] J.L. Guermond, A. Salgado. A splitting method for incompressible ﬂows with variable density based on a pressure Poisson equation. *J.*

*Comput. Phys.*, 2009.

[37] X. He, M. Neytcheva, S. Serra Capizzano. On an augmented

Lagrangian-based preconditioning of Oseen type problems. Department of Information Technology, Uppsala University, Sweden. TR

2010-026, 2010.

39

and global existence of solutions of two-dimensional ﬂows of ﬂuids with pressure- and shear-dependent viscosities. *Math. Comput. Si-*

*mulation*, 61:297-315, 2003.

[39] D. Jacqmin. Contact-line dynamics of a diﬀuse ﬂuid interface. *J.*

*Fluid Mech*, 402:57-88, 2000.

[40] D. Kay, D. Loghin, A.J. Wathen. A preconditioner for the steadystate Navier-Stokes equations. *SIAM J. Sci. Comput.*, 24:237-256,

2002.

[41] J. Kraus. Algebraic multilevel preconditioning of ﬁnite element matrices using local Schur complements. *Num. Lin. Alg. Appl.*, 13:49-70,

2006.

[42] L.Yu. Kolotilina, A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory.

*SIAM J. Matrix Anal. Appl.*,

14:45–58, 1993.

ﬂows of ﬂuids with pressure and shear dependent viscosities. *Appl.*

*Math. Lett.*, 15:961-967, 2002.

[44] M. Neytcheva, M. Do-Quang, X. He. Element-by-element Schur complete approximations for general nonsymmetric matrices of twoby-two block form. *Lecture Notes in Computer Science (LNCS)*,

108:108-115, 2010.

[45] A. Novick-Cohen. The Cahn-Hilliard equation: mathematical and modelling perspectives. *Adv. Math. Sci. Appl.*, 8:965-985, 1998.

[46] A. Novick-Cohen. *The Cahn-Hilliard equation: Handbook of Diffe-*

*rential equations*. IV Evolutionary Partial Diﬀerential Equations, Editors: C. Dafermos and M. Pokorny, Elsevier 2008.

[47] A. Onuki, K. Kawasaki. Nonequilibrium steady state of critical ﬂuids under shear ﬂow: a renormalization group approach. *Ann. Phys.*,

121:456-528, 1979.

[48] P. Kunkel, V. Mehrmann. *Differential-Algebraic Equations: Analysis*

*and Numerical solution*. European Mathematical Society Textbook in Mathmatics, 2006.

[49] M.A. Olshanskii, Y. Vassilevski. Pressure Schur complement preconditioners for the discrete Oseen problem. *SIAM J. Sci. Comput.*,

29:2686-2704, 2007.

40

[50] A. Ouazzi, S. Turek, J. Hron. Finite element methods for the simulation of incompressible powder ﬂow. *Comm. Numer. Methods*

*Engrg.*, 21:581-596, 2005.

[51] T.C. Papanastasiou. Flows of materials with yield, *J. Rheol.*, 31:385-

404, 1987.

[52] Jae-Hong Pyo, J. Shen. Gauge-Uzawa methods for incompressible ﬂows with variable density. *J. Comput. Phys.*, 221:181-197, 2007.

[53] P. Seppecher. Moving contact lines in the Cahn-Hilliard theory. *Int.*

*J. Eng. Sci.*, 34(9):977-92, 1996.

[54] L. de Sobrino, J. Peternelj. Note on capillary waves in the gradient theory of interfaces. *Can. J. Phys.*, 63:1132-33, 1985.

*Mech. Anal.*, 33:377-385, 1969.

demic Press, London, 2001.

[57] J.D. van der Waals. Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering, *Verhandlingen*

*der Koninglijke Akademie van Wetenschappen te Amsterdam, Sec. 1*

(Dutch; English translation in *J. Stat. Phys.*, 1979).

[58] A.J. Wathen. Realistic eigenvalue bounds for the Galerkin mass matrix. *IMA J. Numer. Anal.*, 7:449-457, 1987.

[59] W. Villanueva, G. Amberg. Some generic capillary-driven ﬂows. *Int.*

*J. Multiphase Flow*, 32:1072-1086, 2006.

41

Recent licentiate theses from the Department of Information Technology

2011-001 David Ekl¨ov: *Efficient Methods for Application Performance Analysis*

2010-005 Mikael Laaksoharju: *Let Us Be Philosophers! Computerized Support for Eth-*

*ical Decision Making*

2010-004 Kenneth Duru: *Perfectly Matched Layers for Second Order Wave Equations*

2010-003 Salman Zubair Toor: *Managing Applications and Data in Distributed Comput-*

*ing Infrastructures*

2010-002 Carl Nettelblad: *Using Markov Models and a Stochastic Lipschitz Condition*

*for Genetic Analyses*

2010-001 Anna Nissen:

*Absorbing Boundary Techniques for the Time-dependent*

*Schr¨odinger Equation*

2009-005 Martin Kronbichler: *Numerical Methods for the Navier-Stokes Equations Ap-*

*plied to Turbulent Flow and to Multi-Phase Flow*

2009-004 Katharina Kormann: *Numerical Methods for Quantum Molecular Dynamics*

2009-003

Marta L´arusd´ottir: *Listen to Your Users - The Effect of Usability Evaluation on*

*Software Development Practice*

2009-002 Elina Eriksson: *Making Sense of Usability - Organizational Change and Sense-*

*making when Introducing User-Centred Systems Design in Public Authorities*

2009-001 Joakim Eriksson: *Detailed Simulation of Heterogeneous Wireless Sensor Net-*

*works*

2008-003

Andreas Hellander: *Numerical Simulation of Well Stirred Biochemical Reac-*

*tion Networks Governed by the Master Equation*

2008-002 Ioana Rodhe: *Query Authentication and Data Confidentiality in Wireless Sen-*

*sor Networks*

2008-001 Mattias Wiggberg:

*Unwinding Processes in Computer Science Student*

*Projects*

2007-006 Bj¨orn Halvarsson: *Interaction Analysis and Control of Bioreactors for Nitro-*

*gen Removal*

2007-005 Mahen Jayawardena: *Parallel Algorithms and Implementations for Genetic*

*Analysis of Quantitative Traits*

2007-004 Olof Rensfelt: *Tools and Methods for Evaluation of Overlay Networks*

2007-003

Thabotharan Kathiravelu: *Towards Content Distribution in Opportunistic Net-*

*works*

2007-002 Jonas Boustedt: *Students Working with a Large Software System: Experiences*

*and Understandings*

Department of Information Technology, Uppsala University, Sweden

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project