Fourier Analysis and Local Fourier Analysis for …...Abstract The aim of this master thesis is to...

JO H A NN E S K EP L E RUN I V ER S I T AT L I NZ

Ne t zw e r k f u r F o r s c h u n g , L e h r e u n d P r a x i s

Fourier Analysis and Local Fourier Analysis forMultigrid Methods

Master’s Thesis

for obtaining the academic title

Master of Science

in

INTERNATIONALER UNIVERSITASLEHRGANG

INFORMATICS: ENGINEERING & MANAGEMENT

composed at ISI-Hagenberg

Handed in by:

Yao Zhou, 0856068

Finished on:

July 9th, 2009

Scientific Advisor:

O.Univ. Prof. Dipl. Ing. Dr. Ulrich. Langer.

Hagenberg, July 2009

Johannes Kepler UniversitatA-4040 Linz · Altenbergerstraße 69 · Internet: http://www.jku.at · DVR 0093696

Abstract

The aim of this master thesis is to apply Fourier analysis and local Fourier analysis(LFA) to calculate the exact convergence factor of two-grid methods for the solutionof linear algebraic systems arising in different applications. We first consider the one-dimensional Poisson problem. Both Fourier analysis and LFA are used to derive two-grid convergence factors. Differences between these two analysis tools are illustrated.Then we study a saddle-point problem stemming from an optimal control problem.The smoothing factor for the optimal control problem is derived by Fourier analysisfor a special preconditioner. Moreover, the robustness of the preconditioner is shownby the numerical experiment. In the conclusion we summarize the results and give anoutlook to further investigation. In particular, we are interested in the Fourier analysisfor a mixed formulation of the biharmonic equation that leads to a system of finiteelement equations that is quite similar to the system arising from the optimal controlproblem.

i

Acknowledgments

First of all I would like to express my deep gratitude to Prof. Bruno Buchberger forgiving me this great opportunity to study at the International School for Informaticsand many thanks for all he tried and did for us. What I have learned from ISI andProf. Buchberger is a precious experience that will inspire me for all the rest of my life.I would like to thank all those who were actively involved in coordinating the activitiesof ISI, especially Betina Curtis, making my one year study in Austria enjoyable andcomfortable.

I am very grateful to my supervisor Prof. Ulrich Langer, for guiding and encourag-ing me during my work on the master thesis. I would also like to thank Prof. WalterZulehner and Dr. Georg Regensburger for discussions on my work.

In particular, I want to thank my friend Dr. Xiliang Lu for his valuable contribu-tions and strong help.

I want to acknowledge Johann Radon Institute for Computational and AppliedMathematics (RICAM) for offering me this chance to writing my master’s thesis andfor supporting my studies and stay in Austria.

I profoundly thank and appreciate very much the efforts of my parents, my grand-parents and my boyfriend, who always supported me in my choices. I am grateful tomy college Le Wang who encouraged me a lot and other friends who also participatein the ISI2008 program for sharing the difficulties and happiness with me.

Yao ZhouLinz, July 2009

iii

Contents

1 Introduction 1

2 Multigrid Methods 52.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Convergence Analysis for the ω-Jacobi Iteration . . . . . . . . . 62.1.3 Smoothing Properties of ω-Jacobi Relaxation . . . . . . . . . . 7

2.2 Two-Grid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Fourier Analysis and Local Fourier Analysis 153.1 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Asymptotic Two-Grid Convergence . . . . . . . . . . . . . . . . 153.1.2 Towards Local Fourier Analysis . . . . . . . . . . . . . . . . . . 21

3.2 Local Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Formal Tools of Local Fourier Analysis . . . . . . . . . . . . . . 223.2.2 Smoothing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Two-grid Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Application to an Optimal Control Problem 294.1 Fourier analysis for Smoothing Operator . . . . . . . . . . . . . . . . . 31

4.1.1 Smoothing Operator . . . . . . . . . . . . . . . . . . . . . . . . 314.1.2 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . . 314.1.3 Smoothing Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Fourier Two-grid Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Local Fourier Analysis for Smoothing Operator . . . . . . . . . . . . . 374.4 Local Fourier Two-grid Analysis . . . . . . . . . . . . . . . . . . . . . . 384.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Conclusion 43

v

vi CONTENTS

Chapter 1

Introduction

The main goal of this work is to apply Fourier analysis and local Fourier analysis(LFA) to calculate the convergence rate of multigrid methods for solving linear sys-tem of equations coming from the finite element discretization of partial differentialequations.

The discretization of partial differential equations leads to very large systems ofequations. For instance, in three space dimensions, more than one million unknownscan be reached very easily. The direct solution of systems of this size is prohibitivelyexpensive, both with respect to the amount of storage and to the computational work.Therefore iterative methods like the Gauss-Seidel or the Jacobi iteration have beenwidely used from the beginning of the numerical treatment of partial differential equa-tions.

An important step was Young’s successive over-relaxation method [28] (1950) whichis much faster than the closely related Gauss-Seidel iteration. Nevertheless, thismethod shares with direct elimination methods the disadvantage that the amountof work does not remain proportional to the number of unknowns; the computer timeneeded to solve a problem grows more rapidly than the size of the problem.

Multigrid methods were the first to overcome this complexity barrier. Multigridalgorithms are composed of four elements:

(1) smoothing of rough error parts via classical iterative methods;

(2) approximation of smooth errors on a coarser grid;

(3) recursive application of (1) and (2) on a sequence of coarser and coarser grids;

(4) nested iteration for producing good initial guesses.

In 1961, Fedorenko proposed the first two-grid method (elements (1) and (2)) inhis pioneering paper [11] for solving the finite difference equations arising from the

1

2 CHAPTER 1. INTRODUCTION

five-point approximation of the Poisson equation in a rectangular domain. Alreadythe first numerical results presented in this paper showed the great potential of thismethod. In his second paper [12] from 1964, he gave a first convergence analysisand proposed to combine (1) and (2) with (3) that results in a complete multigridcycle. In 1966, Bakhvalov provided the first rigorous convergence proof on the basisof the sum splitting technique and showed that the nested iteration technique withthe multigrid as nested iteration process allows us to produce approximate solutionsthat differ from the exact solution of the boundary value problem in the order ofthe discretization error with optimal complexity [2]. In the 1970s, these techniqueswere generalized to variational finite difference equations and general finite elementequations by Astrakhantsev [1] and by Korneev [20], respectively.

The paper of Brandt [6] and the report of Hackbusch [13] were the historical breakthrough. They reinvented the multigrid method independently of the Russian school,generalized it to new classes of problems, and developed the theory. In his first two pa-pers [4, 5] (1973, 1976) and then summarized in the systematic 1977 work [6], Brandtshowed the actual efficiency of multigrid methods. His essential contributions (in theearly studies) include the introduction of a nonlinear multigrid method (FAS) and ofadaptive techniques (MLAT), the discussion of general domains, the systematic appli-cation of the nested iteration idea (FMG) and, last but not least, the introduction ofthe Local Fourier Analysis as a tool for theoretical investigation and method design.Hackbusch’s first systematic report [13] (1976) contains many theoretical and practi-cal investigations such as Fourier analysis considerations which have been taken upand developed further by several authors. In the papers [14, 15, 16], Hackbusch thenpresented a general convergence theory of multigrid methods. The first big multigridconference in 1981 in Cologne was a culmination point of the development; the confer-ence proceedings edited by Hackbusch and Trottenberg [17] are still a basic reference.The first monograph on multigrid methods was published by Hackbusch in 1985 [18].In this monograph, the reader finds a complete overview of the most important resultson multigrid and related methods obtained till 1985.

Today, multigrid methods are used in nearly every field where partial differentialequations are solved by numerical methods. After the pioneering monograph [18] byHackbusch, many further books have been published wherein one can find interestingadditional information on the theory and application of multigrid methods in surveyform, e.g. the books by Briggs, Henson and McCormick [10], Bramble [3], Wesseling[26], Trottenberg, Oosterlee and Schuller [25]. The most recent information aboutpublications on multigrid method, related conferences, multigrid software, etc., can befound on the multigrid home page http://www.mgnet.org on the internet.

Considering multigrid algorithms, there is an enormous degree of freedom in choos-ing the algorithmic components. The practical very important question is how tochoose individual multigrid components for concrete situations. There are two relatedapproaches in treating this question: Fourier analysis and local Fourier analysis (LFA).They are considered as the main analysis tools to obtain quantitative convergence es-

http://www.mgnet.org

3

timates and to optimize multigrid components like smoothers or intergrid transferoperators.

In Fourier analysis and LFA, the discrete solution and the corresponding error canbe represented by linear combinations of certain functions, which form a unitary basisof the space of grid functions with bounded l2-norm. However, in Fourier analysiswe use the exact eigenfunctions of the discrete operator which are compatible withthe boundary conditions. Whereas in LFA an infinite grid is considered and bound-ary conditions are not taken into account. In this case, we use exponential functionsinstead of the exact eigenfunctions. Both ideas were introduced by Brandt [6] andafterward extended and refined in [7, 9]. A good introduction can be found in thepaper by Stuben and Trottenberg [21] and in the books by Wesseling [26], Trottenberget al. [25] and Wienands and Joppich [27]. In particular, the book by Wienands andJoppich [27] provides with the software package LFA for local Fourier Analysis.

The paper is organized as follows: In Chapter 2 we introduce the basic ideas of multi-grid methods and illustrate the multigrid algorithm. Chapter 3 serves as a demonstra-tion of how model problem can be investigated by means of Fourier analysis and LFA.Connections and differences of these two analysis tools are described. The main partof the paper is the application to PDE-constrained optimization problem in Chapter4, both Fourier analysis and LFA are applied to compute the two-grid convergencerate and also numerical results are presented. In Chapter 5 we summarize the resultsand conclude with some remarks about possible continuation.

4 CHAPTER 1. INTRODUCTION

Chapter 2

Multigrid Methods

Multigrid methods are among the fastest numerical algorithms for solving linear sys-tems arising from a discretization technique such as the finite element method or thefinite difference method. The convergence speed does not deteriorate when the dis-cretization is refined, whereas classical iterative methods slow down for decreasingmesh size. For a wide overview about multigrid methods we recommend the booksHackbusch and Trottenberg [19], Hackbusch [18], Bramble [3] and Trottenberg, Oost-erlee and Schuller [25].

The starting point of the multigrid idea is the observation that classical iterationmethods (like the Gauss-Seidel iteration and the damped Jacobi method) have smooth-ing properties. Although these methods are characterized by poor global convergencerates, for errors whose length scales are comparable to the mesh size, they provide rapiddamping, leaving behind smooth, longer wave-length errors. These smooth parts ofthe error are responsible for the poor convergence. A (geometric) multigrid methodinvolves a hierarchy of meshes and related discretizations. A quantity that is smoothon a certain grid can, without any essential loss of information, also be approximatedon a coarser grid. So the low-frequency error components can be effectively reduced bya coarse-grid correction procedure. Because the action of a smoothing iteration leavesonly smooth error components, it is possible to represent them as the solution of anappropriate coarser system. Once this coarser problem is solved, its solution is interpo-lated back to the fine grid to correct the fine grid approximation for its low-frequencyerrors.

Thus, the multigrid idea is based on two principles: error smoothing and coarse gridcorrection. In this chapter, we will explain how these principles are combined to form amultigrid algorithm. The basics of multigrid methods will be described systematically.

In Section 2.1 , we discuss the smoothing properties of one classical iterative solver,typically the Jacobi method. The concrete example refers to Poisson’s equation. Sec-tions 2.2 and 2.3 give a systematic introduction to two-grid iteration and multigriditeration, respectively.

5

6 CHAPTER 2. MULTIGRID METHODS

2.1 Motivation

2.1.1 Model Problem

A natural model of an elliptic equation is the two-dimensional Poisson equation in asquare. However, the analysis of classical iterative methods as well as the multigridalgorithm for this problem is not sufficiently transparent for an introductory consider-ation. Therefore, we start with the simplest one-dimensional Dirichlet boundary valueproblem

− u(x)′′ = f(x) in Ω = (0, 1), u(x) = 0 at x ∈ Γ = 0, 1. (2.1)

Discretising the model problem by finite element method with piecewise linear, con-tinuous functions we obtain a linear system of equations

Khuh = fh, (2.2)

with the tridiagonal matrix

Kh =1

h

2 −1−1 2 −1

−1. . . . . .. . . . . . −1

−1 2

(n−1)×(n−1)

(2.3)

and the vectorsuh = (uh(jh))n−1

j=1 , fh

= (fh(jh))n−1j=1 .

The boundary conditions uh(0) = uh(1) = 0 are incorporated into the system.Here h denotes the grid size. The grid Ωh corresponding to h = 1/n consists of the

points xj = jh (j = 1, . . . , n− 1). From the beginning we introduce not only one gridbut a hierarchy of grids. The largest possible grid size is h0 = 1/2. The correspondinggrid Ω0 consists of the single grid point x1 = 1/2. The next finer grid should be definedby halving the step size (standard coarsening): h1 = 1

4. Thus, this refinement process

yields the sequence

h0 > h1 > h2 . . . > hl . . . with hl = 2−1−l. (2.4)

The subscript l is called the level number. In this paper, we assume that n is an evennumber and choose ΩH with mesh size H = 2h to be the coarse grid.

2.1.2 Convergence Analysis for the ω-Jacobi Iteration

The asymptotic convergence property of an iterative method is characterized by thespectral radius of its corresponding operator

ρ(Sh) = max|λ| : λ is eigenvalue of Sh

2.1. MOTIVATION 7

which is also called the asymptotic convergence factor. The iterative process convergesif and only if ρ(Sh) < 1.

For model problem (2.1), the ω-Jacobi method reads

uj+1h = uj

h +ωh

2(f

h−Khu

jh). (2.5)

Clearly, the corresponding iteration operator is given by

Sh = Sh(ω) = I − ωh

2Kh. (2.6)

The spectral radius of the iteration operator for the Jacobi method can be deter-mined analytically for our model problem. The first step is to verify that

ϕkh = (

√2h sin(kπxj))

n−1j=1 = (

√2h sin(kjπh))n−1

j=1 k = 1, ..., n− 1. (2.7)

and

λkh =

4

hsin2(

1

2kπh) (2.8)

are eigenfunctions, respectively eigenvalues of Kh.The eigenfunctions of Sh are the same as those of Kh. The corresponding eigenvalues

of Sh are

λkh(Sh) = 1− ωh

2λk

h = 1− 2ω sin2(1

2kπh) = 1− ω + ω cos kπh. (2.9)

For the spectral radius ρ(Sh) = ρ(h, ω) = max|λkh(ω)| : k = 1, . . . , n− 1, we obtain

for 0 < ω ≤ 1 : ρ(Sh(ω)) = |λ1h(ω)| = 1− ω + ω cos πh = 1−O(h2),

else : ρ(Sh) ≥ 1 (for h small enough). (2.10)

Thus, the choice ω ∈ (0, 1] implies convergence. However, the spectral radius is veryclose to one if h is small and there is no satisfactory convergence for any ω. Wecan easily spot the h-dependent convergence: the smaller h becomes the worse theconvergence is.

2.1.3 Smoothing Properties of ω-Jacobi Relaxation

The situation is quite different with respect to the smoothing properties of ω-Jacobirelaxation. For 0 < ω ≤ 1, we first observe by (2.10) that it is the smoothest eigen-function ϕ1

h which is responsible for the slow convergence of Jacobi’s method. Highlyoscillating eigenfunctions are reduced much faster if ω is chosen properly. The con-struction of the multigrid iteration is based on this observation.

To see this, we consider the approximations before (wh) and after (wh) one relax-ation step. Since the eigenfunctions ϕk

h of the iteration operator form a basis for all


grid functions having zero boundary values, we can expand the errors before and afterone relaxation step, namely

vh := uh − wh and vh := uh − wh

into discrete eigenfunction series:

vh =n−1∑

k=1

αkϕkh, vh =

n−1∑

k=1

λkh(Sh(ω))αkϕ

kh (2.11)

In order to analyze the smoothing properties of Sh(ω) we distinguish low and highfrequencies (with respect to the coarser grid ΩH used). For the definition of the highand low frequencies, we return to the eigenfunctions ϕk

h in (2.7). For given k, weconsider the eigenfunctions

ϕkh, ϕn−k

h

and observe that they coincide on ΩH in the following sense:

ϕkh(xj) = −ϕn−k

h (xj) for xj ∈ ΩH . (2.12)

This means that these two eigenfunctions cannot be distinguished on ΩH . (For k =n/2, the ϕk

h vanishes on ΩH .) This gives rise to the following definition of low andhigh frequencies:

Definition 2.1. (in the context of model problem)

low frequencies : ϕkh with k < n/2,

high frequencies : ϕkh with n/2 ≤ k < n. (2.13)

Obviously, only the low frequencies are representable on the coarser grid ΩH . Thehigh frequencies are not visible on ΩH since either they coincide with a low frequencyon ΩH or vanish on ΩH .

In order to measure the smoothing properties of Sh(ω) quantitatively, we now in-troduce the smoothing factor µ(h; ω) of Sh(ω) (and its supremum µ∗(ω) over h) asfollows:

Definition 2.2. Smoothing factor (of ω-Jacobi for model problem)The smoothing factor µ(h; ω) of Sh(ω), representing the worst factor by which highfrequency error components are reduced per relaxation step, and its supremum µ∗ overh, are defined as

µ(h; ω) := max|λkh(Sh(ω))| : n/2 < k ≤ n− 1,

µ∗(ω) := supµ(h; ω) : h ≤ 1/2 (2.14)

2.2. TWO-GRID METHOD 9

Inserting (2.9), we obtain from (2.14)

µ(h; ω) = max|1− ω|, |1− ω(1 + cos πh)|,µ∗(ω) = max|1− ω|, |1− 2ω| (2.15)

This shows that Jacobi’s relaxation has no smoothing properties for ω ≤ 0 or ω > 1:

µ(h; ω) ≥ 1 if ω ≤ 0 or ω > 1 (and h sufficiently small)

For 0 < ω < 1, however, the smoothing factor is smaller than 1 and bounded awayfrom 1, independently of h. For ω = 1, we have a smoothing factor of 1−O(h2) only.In particular, we find from (2.15) that

µ(h; ω) =

cos πh if ω = 11/2 if ω = 1/21/3 if ω = 2/3

µ∗(ω) =

1 if ω = 11/2 if ω = 1/21/3 if ω = 2/3

The choice ω = 2/3 is optimal in the following sense:

infµ∗(ω) : 0 ≤ ω ≤ 1 = µ∗(2

3) =

1

3. (2.16)

With respect to µ(h; ω), we obtain

infµ(h; ω) : 0 ≤ ω ≤ 1 = µ(h;2

2 + cos πh) =

cos πh

2 + cos πh=

1

3− |O(h2)|.

This means that one step of ω-Jacobi with ω = 2/3 reduces all high frequency errorcomponents by at least a factor of 1/3 (independent of the grid size h).

The above consideration is an example of what we call smoothing analysis.

2.2 Two-Grid Method

The foregoing section showed that the simple ω-Jacobi iteration (2.5) with a properlychosen ω is quite an efficient method for reducing the high-frequency components.Convergence is only lacking with respect to low frequencies (smooth components).Therefore, one should combine this iteration with a second one having complementaryproperties. In particular, the second iteration should reduce the smooth error verywell. Such a complementary iteration can be constructed by means of the coarse gridwith step size hl−1 = 2hl. For our model problem, we use H = 2h.

Let uoldh be some given approximation to uh = Khfh

. A small number of iterationsof method (2.5) will result in an intermediate value uh. From the previous section weknow that the error vh = uh− uh is smooth (more precisely, smoother than uh−uold

h ).vh can also be regarded as the exact correction since uh = uh + vh.

Inserting uh into the equation Khuh = fh

we obtain the defect

dh = fh−Khuh (2.17)


of uh, which vanishes if and only if uh is the exact solution uh. Because of Khvh =Khuh −Khuh = Khuh − f

hthe exact correction vh is the solution of

Khvh = dh. (2.18)

The defect equation (2.18) is equivalent to the original equation (2.2). The exactsolution of the defect equation is as difficult as for the original one. Nevertheless, vh

can be approximated better by means of a coarser grid than uh, since vh is a smoothgrid function and only smooth functions can be represented well by means of coarsergrids.

To approximate the problem (2.18) by a coarse grid equation

KHvH = dH (2.19)

we have to choose dH reasonably. Note that the matrix KH is already defined by theunderlying bilinear form and the basis functions of the coarse finite dimensional spaceVH . We define dH as the restriction of the fine grid defect to the coarse grid, that is

dH = IHh dh (2.20)

Here IHh is called the restriction. For our model problem we use the weighted restriction

given by

IHh dh(x) =

1

2dh(x− h) + dh(x) +

1

2dh(x + h), for x ∈ ΩH (2.21)

The corresponding matrix is

IHh =

1

2

1 2 1 O1 2 1

. . .

1 2 1O 1 2 1

(n2−1)×(n−1)

(2.22)

Having defined dH by (2.20) from the defect dh, we obtain the exact solution vH ofthe coarse grid equation (2.19). Reasonably one expects vH to be an approximationto the exact correction vh. However vH is only defined on the coarse grid ΩH . Wehave to interpolate this coarse grid function by

vh = IhHvH , (2.23)

where IhH is called the prolongation. The simplest example for a prolongation operator

is the linear interpolation

IhHvH(x) =

vH(x) if x ∈ ΩH12[vH(x− h) + vH(x + h)] otherwise.

(2.24)

2.2. TWO-GRID METHOD 11

The corresponding matrix is

IhH =1

2

121 1

2. . .

21 1

21

(n−1)×(n2−1)

(2.25)

Since uh = u + vh is the exact solution and vh = IhHvH is supposed to approximate

vh, we update the value uh byunew

h = uh + vh. (2.26)

All steps from uh to unewh by (2.18)-(2.26) can be combined into the so-called coarse

grid correction stepunew

h = uh − IhHK−1

H IHh (Khuh − f

h). (2.27)

The associated iteration operator is given by

MCGCh = I − Ih

HK−1H IH

h Kh. (2.28)

Remark 2.3. Taken on its own, the coarse grid correction process (2.27) is of no use.It is not convergent! We have

ρ(MCGCh ) ≥ 1. (2.29)

This remark follows directly from the fact that the restriction IHh has a nontrivial

kernel. If we choose 0 6= zh ∈ ker(IHh ) and set u0

h := K−1h (f

h− zh). As dh = zh and

zH = IHh zh = 0, the resulting iterates are uj

h = u0h. The solution will not be changed

by a pure coarse grid correction.

It is the combination of smoothing iteration and coarse-grid correction that is rapidlyconvergent, whereas both components by themselves converge slowly or not at all. Thecombination is called a two-grid method since two levels h and H are involved. Eachiteration of a two-grid method consists of a smoothing and a coarse grid correction part.


One step of such a two-grid method (calculating uj+1h from uj

h) proceeds as follows:

Two-grid method uj+1h = TGM (ν1,ν2)(uj

h, Kh, fh, ν1, ν2)

(1) Presmoothing

- Compute ujh by applying ν1(≥ 0) steps of a given smoothing method to uj

h:

ujh = Sν1

h (ujh, Kh, fh

)

(2) Coarse grid correction (CGC)

- Compute the defect: djh = f

h−Khu

jh.

- Restrict the defect (fine-to-coarse transfer): djH = IH

h djh.

- Solve exactly on ΩH : KHvH = djH .

- Interpolate the correction (coarse-to-fine transfer): vh = IhHvH

- Compute the corrected approximation: uh = ujh + vh

(3) Postsmoothing

- Compute uj+1h by applying ν2(≥ 0) steps of the given smoothing method to

uh:

uj+1h = Sν2

h (uh, Kh, fh)

From the above description, we immediately obtain the iteration operator MTGMh

of the (h,H) two-grid method:

MTGMh = Sν2

h MCGCh Sν1

h = Sν2h (I − Ih

HK−1H IH

h Kh)Sν1h . (2.30)

2.3 Multigrid Method

Up to now, we have described the multigrid principle only in its two-grid version. How-ever, the two-grid methods usually are not used in practice because the exact solutionof the coarse grid equation is required. The system on the coarse grid is easier to solvethan the one on the fine grid. By choosing a coarse grid size H = 2h and uniformrefinement, the number of unknowns decreases about to an half (in one dimension).In practical applications we are interested in solving fine grid equations with millionsof unknowns, therefore, the coarse grid equation has still a too large complexity, suchthat the overall two-grid method is not optimal with respect to the computationaleffort. They serve only as the (theoretical) basis for the real multigrid method.

2.3. MULTIGRID METHOD 13

The multigrid idea starts from the observation that in a convergent two-grid methodit is not necessary to solve the coarse grid defect equation (2.19) exactly. Instead,without essential loss of convergence speed, one may replace vH by a suitable approx-imation. A natural way to obtain such an approximation is to apply an analogoustwo-grid method to (2.19) also, where an even coarser grid than ΩH is used. Clearly,if the convergence factor of this two-grid method is small enough, it is sufficient topreform only a few, say γ iteration steps to obtain a good enough approximation to thesolution of (2.19). This idea can, in a straightforward manner, be applied recursively,using coarser and coarser grids, down to some coarsest grid. On this coarsest grid anysolution method can be used (e.g. a direct method or an iterative method if it hassufficiently good convergence properties on the coarsest grid).

The recursive multigrid method (MGM) on level l is then given by the algorithmbelow. The sequence of operations during one step of the multigrid iteration for γ = 1and γ = 2 is depicted in Figure 2.1. Here , ¤, and mean smoothing, solvingexactly, fine-to-coarse and coarse-to-fine transfer, respectively. Due to the Figure 2.1the cases with γ = 1 and γ = 2 are called V-cycle and W-cycle respectively.

Multigrid method MGM(l, u, f)if l = 0 then

u := K−10 f // exact solve on coarsest level

else

u := Sν1l (u, f); //presmoothing step

d := I l−1l (Klu− f); //restriction of the defect

v := 0; //starting valuefor j := 1, . . . , γ, do

MGM(l− 1, v, d) // solve Kl−1vl−1 = dl−1 by applying γ step of MGM(l− 1, v, d)end for

u := u− I ll−1v //update on level l

u := Sν2l (u, f) //postsmoothing step

end if

Figure 2.1: V-cycle and W-cycle

Chapter 3

Fourier Analysis and Local FourierAnalysis

In this chapter, we show how rigorous Fourier analysis and local Fourier analysis(LFA) can be used to derive two-grid convergence factors. The purpose of this chapteris two-fold: first, we want to introduce the idea of rigorous Fourier analysis and LFAand to explain their structures, respectively. Secondly, we want to compute the exacttwo-grid convergence rate for one-dimensional model problem first by rigorous Fourieranalysis and then by LFA.

3.1 Fourier Analysis

3.1.1 Asymptotic Two-Grid Convergence

In the motivation of multigrid method in Section 2.1, we have already studied thesmoothing rate of ω-Jacobi iteration using some facts that are characteristic for therigorous Fourier analysis. We again start with ϕk

h (k = 1, . . . , n − 1) from (2.7), thediscrete eigenfunctions of Kh. The vectors ϕk

h : 1 ≤ k ≤ n− 1 form an orthonormalbasis of Rn−1, since

(ϕkh, ϕ

lh)l2 =

n−1∑j=1

√2h sin(kπjh)

√2h sin(lπjh) = δk,l 1 ≤ k, l ≤ n− 1

Therefore,the matrix Qh built by ϕkh as columns is unitary: Q−1

h = QTh ,

Qh = [ϕ1h, ϕ

n−1h , ϕ2

h, ϕn−2h , . . . , ϕk

h, ϕn−kh , . . . , ϕ

(n/2)−1h , ϕ

(n/2)+1h , ϕ

n/2h ] (3.1)

Since multiplication by Qh or Q−1h does not change the spectral radius, we have

ρ(MTGMh ) = ρ(MTGM

h ) for MTGMh := Q−1

h MTGMh Qh. (3.2)

15

16 CHAPTER 3. FOURIER ANALYSIS AND LOCAL FOURIER ANALYSIS

We will show that the Fourier-transformed iteration matrix Q−1h MTGM

h Qh of the two-grid method is a block-diagonal matrix of the form

MTGMh =

M(1)h

M(2)h

. . .

M(n/2−1)h

M(n/2)h

(3.3)

with

M(k)h (2× 2)−matrices for 1 ≤ k ≤ n/2− 1, M

(n/2)h (1× 1)−matrix. (3.4)

Thus the determination of the spectral radius ρ(MTGMh ) can be reduced to the calcu-

lation of the spectral radii of (at most) (2× 2)-matrices:

ρ(MTGMh ) = ρ(MTGM

h ) = maxρ(M(k)h ) : 1 ≤ k ≤ n/2. (3.5)

For the proof of the block structure (3.3)-(3.4), we transform the iteration matrix

MTGMh = Sν2

h MCGCh Sν1

h with Sh = I − ωh

2Kh, MCGC

h = I − IhHK−1

H IHh Kh. (3.6)

into

MTGMh = Sν2

h (I − IhHK−1

H IHh Kh)S

ν1h with Kh := Q−1

h KhQh, Sh := Q−1h ShQh. (3.7)

To define the quantities

IhH := Q−1

h IhHQH , KH := Q−1

H KHQH , IHh := Q−1

H IHh Qh, (3.8)

we have to introduce the Fourier transformation QH at level H. Replacing h in (2.7)by H = 2h, we obtain the eigenfunctions of KH :

ϕkH = (

√4h sin(2kjπh))

n/2−1j=1 k = 1, ..., n/2− 1. (3.9)

Analogously to (3.1), the vectors ϕkH form the matrix

QH = [ϕ1H , ϕ2

H , . . . , ϕn/2−1H ]. (3.10)

According to (2.7)-( 2.8), Khϕkh = λk

hϕkh holds with λk

h = 4h

sin2(12kπh). We introduce

s2k = sin2(

1

2kπh), c2

k = cos2(1

2kπh). (3.11)

3.1. FOURIER ANALYSIS 17

Noting that λn−kh = 4

hs2

n−k = 4hc2k, we obtain

Kh := Q−1h KhQh =

K(1)h

K(2)h

. . .

K(n/2)h

(3.12)

with the blocks

K(k)h =

4

h

[s2

k 00 c2

k

]for 1 ≤ k ≤ n/2− 1, K

(n/2)h =

2

h. (3.13)

For Sh we have

Shϕkh = (1− 2ωs2

k)ϕkh,

Shϕn−kh = (1− 2ωc2

k)ϕn−kh .

Thus, we obtain

Sh := Q−1h ShQh =

S(1)h

S(2)h

. . .

S(n/2)h

(3.14)

with the blocks

S(k)h =

[1− 2ωs2

k 00 1− 2ωc2

k

]for 1 ≤ k ≤ n/2− 1, S

(n/2)h = 1− ω. (3.15)

Since KHϕkH = λk

HϕkH with λk

H = 42h

sin2(12kπ2h) = 2

hsin2(kπh) and sin2(kπh) = 4s2

kc2k,

we obtain the diagonal matrix

KH := Q−1H KHQH = diagK(1)

H , . . . , K(n/2−1)H with K

(k)H =

8

hs2

kc2k. (3.16)

Next, we study the transfer operators IHh and Ih

H defined by (2.21) and (2.24) respec-tively. From the definition of IH

h :

IHh uh(x) =

1

2uh(x− h) + uh(x) +

1

2uh(x + h), for x ∈ ΩH

we get

IHh sin(kπx) = [sin(kπ(x− h)) + 2 sin(kπx) + sin(kπ(x + h))]/2

= [1 + cos(kπh)] sin(kπx)

= 2 cos2(1

2kπh) sin(kπx)

= 2c2k sin(kπx) for x ∈ ΩH


Thus,IHh ϕk

h =√

2c2kϕ

kH . (3.17)

Since this identity holds for all k, we replace k by n− k:

IHh ϕn−k

h =√

2c2n−kϕ

n−kH =

√2s2

kϕn−kH .

For 0 ≤ k ≤ n/2, the equality sin(2kjπh) = − sin(2(n− k)jπh) leads to

ϕn−kH = −ϕk

H .

Then, we getIHh ϕn−k

h = −√

2s2kϕ

kH . (3.18)

From (3.17) and (3.18) we obtain the representation

IHh := Q−1

H IHh Qh =

r(1) 0r(2) 0

. . ....

r(n/2−1) 0

(3.19)

withr(k) =

√2[

c2k, −s2

k

]. (3.20)

This means that the last column of IHh vanishes (this follows from IH

h ϕn/2h = 0) and

that the remaining part of the format (n/2− 1)× (n− 2) consists of n/2− 1 (1× 2)-blocks r(k).

IhH from (2.21) and IH

h from (2.24) are connected by IhH = (IH

h )T , thus, we derivethe representation

IhH := Q−1

h IhHQH = Q−1

h (IHh )T QH = QT

h (IHh )T QH = (QT

HIHh Qh)

T = (IHh )T . (3.21)

From (3.19) we obtain

IhH := Q−1

h IhHQH =

p(1)

p(2)

. . .

p(n/2−1)

0 0 · · · 0

(3.22)

with

p(k) =√

2

[c2k

−s2k

]. (3.23)

Since all factors in (3.6) have a block-diagonal structure, this carries over to MTGMh

and proves the structure (3.3)-( 3.4). For the (2 × 2) blocks M(k)h (1 ≤ k ≤ n/2 − 1)

and the 1× 1 block Mn/2h , the relations (3.13), (3.15), (3.16), (3.20) and (3.23) yield


M(k)h = (S

(k)h )ν2(I − p(k)(K

(k)H )−1r(k)K

(k)h )(S

(k)h )ν1 (1 ≤ k ≤ n/2− 1),

M(n/2)h = (1− ω)ν1+ν2 . (3.24)

Inserting the representations of p(k), K(k)H , r(k), K

(k)h and S

(k)h , we obtain

M(k)h =

(S

(k)h

)ν2([

1 00 1

]−√

2

[c2k

s2k

]h

8c2ks

2k

√2[

c2k −s2

k

] 4

h

[s2

k 00 c2

k

]) (S

(k)h

)ν1

=(S

(k)h

)ν2([

1 00 1

]−

[c2k −c2

k

−s2k s2

k

]) (S

(k)h

)ν1

=

[1− 2ωs2

k 00 1− 2ωc2

k

]ν2[

s2k c2

k

s2k c2

k

] [1− 2ωs2

k 00 1− 2ωc2

k

]ν1

. (3.25)

The block M(k)h describes the application of MTGM

h to the two functions ϕkh, ϕ

n−kh

(respective columns of the matrix Qh, cf. (3.1)). Since ρ(AB) = ρ(BA) for any linear

operators A and B, ρ(MTGMh ) = maxρ(M

(k)h ) : 1 ≤ k ≤ n/2 does not depend on ν1

and ν2 individually. It depends in particular on the parameter ω and on ν = ν1 + ν2.In the following, we shall use

ρ(h, ν; ω) := ρ(MTGMh (ν1, ν2; ω)) (3.26)

to denote the convergence factor of two-grid method.Since for k = 1, . . . , n/2− 1,

ρ(M(k)h ) = ρ((S

(k)h )ν2(I − p(k)(K

(k)H )−1r(k)K

(k)h )(S

(k)h )ν1)

= ρ((I − p(k)(K(k)H )−1r(k)K

(k)h )(S

(k)h )ν2(S

(k)h )ν1))

= ρ((I − p(k)(K(k)H )−1r(k)K

(k)h )(S

(k)h )ν)

= ρ

([s2

k(1− 2ωs2k)

ν c2k(1− 2ωc2

k)ν

s2k(1− 2ωs2

k)ν c2

k(1− 2ωc2k)

ν

])

= |s2k(1− 2ωs2

k)ν + c2

k(1− 2ωc2k)

ν |= |s2

k(1− 2ωs2k)

ν + (1− s2k)(1− 2ω + 2ωs2

k)ν |

= ρν(s2k) (3.27)

and

ρν(1

2) = |(1− ω)ν | = ρ(M

(n/2)h )

we getρ(h, ν; ω) = maxρν(s

2k) : 1 ≤ k ≤ n/2. (3.28)

In practice, we are more interested in the asymptotic convergence factor

ρ∗(ν; ω) := supρ(h, ν; ω) : h ≤ 1/2. (3.29)


In our model problem, we have

ρ∗(ν; ω) = supρν(s2k) : 0 < s2

k ≤ 1/2. (3.30)

In particular, we study the optimal choices of ω when ν is fixed to be 1 or 2:

• For ν = 1, we have

ρ1(s2k) = −4ω (s2

k −1

2)2 − ω + 1,

ρ(h; ω) = max|1− ω|, |1− ω(1 + cos2 πh)|,ρ∗(ω) = max|1− ω|, |1− 2ω|.

The choice w = 2/3 is optimal in the following sense:

infρ∗(ω) : 0 ≤ ω ≤ 1 = ρ∗(2

3) =

1

3.

With respect to ρ(h; ω), we obtain

infρ(h; ω) : 0 ≤ ω ≤ 1 = ρ(h;2

2 + cos2 πh) =

cos2 πh

2 + cos2 πh.

• For ν = 2, we obtain from (3.27)

ρ2(s2k) = (−8ω + 12ω2)(s2

k −1

2)2 + (1− ω)2,

ρ(h; ω) = max(1− ω)2, |(3ω2 − 2ω) cos2 πh + (1− ω)2|,ρ∗(ω) = max(1− ω)2, (1− 2ω)2.

The choice w = 2/3 is optimal in the following sense:

infρ∗(ω) : 0 ≤ ω ≤ 1 = ρ∗(2

3) =

1

9.

With respect to ρ(h; ω), we obtain

infρ(h; ω) : 0 ≤ ω ≤ 1 = ρ(h;2

3) =

1

9.

The optimal choices of ω in these two cases are the same with the choice in thesmoothing analysis in (2.16). This is a strong indication that ω = 2/3 is a good choicein the multigrid method.


3.1.2 Towards Local Fourier Analysis

Based on rigorous Fourier analysis many results have been obtained for different modelproblems in 2D and 3D, for different discretizations, coarsening strategies, transfer op-erators and smoothing procedures, e.g. [21, 22, 23].

As we have seen in the previous section, using rigorous Fourier analysis for an exactcalculation of the spectral radius of the k-grid operator which are quantitative mea-sures for the error reduction, it is necessary that there exists a unitary basis of periodiceigenfunctions of Kh which generate the whole space of grid functions and which arecompatible with the boundary conditions. Then it is possible to expand the error afterthe i-th k-grid iteration into a Fourier series by a unitary basis transformation, anddifferent multigrid methods can be analyzed by evaluating their effect on the eigen-functions. The main disadvantages of rigorous Fourier analysis is, however, that it candirectly be applied to a rather small class of problems. As soon as we deal with morecomplicated domains, operators or boundary conditions an appropriate basis of uni-tary eigenfunctions cannot be derived explicitly. Thus, we will, in general, not be ableto apply the rigorous Fourier analysis. The situation is different for the local Fourieranalysis (LFA). We will introduce the basic ideas and formalism of LFA in next section.

In this section, we will point out that the rigorous Fourier analysis and the LFA areclosely related. The mathematical substance of both is very similar.

The relationship is most easily explained by reconsidering the 1D Poisson equationwith periodic boundary conditions. Using again finite element discretization and forthe corresponding grid Ωh, the discrete eigenfunctions are

ϕkh = ei2πkx (k = 0, 1, . . . , n− 1). (3.31)

Because of periodicity, we have

ei2πk′x = ei2πkx on Ωh if k′ = k (mod n). (3.32)

Shifting k and assuming n to be even, we can thus renumber the eigenfunctions (3.31)as follows

0 ≤ k < n− 1 ←→ −n

2≤ k <

n

2. (3.33)

This numeration is convenient and is customary in the local Fourier analysis contextdescribed in the next section. With the notation

θ := 2πk

n= 2πkh (3.34)

and

Θh = θ = 2πk

n: −n

2≤ k <

n

2; k ∈ Z, (3.35)

we can write the discrete eigenfunctions (3.31) in the form

ϕh(θ, x) = eiθx/h (θ ∈ Θh). (3.36)


Obviously,Θh ⊂ [−π, π), #Θh = n. (3.37)

In case of standard coarsening we can split Θh into subsets corresponding to low andhigh frequencies

Θlowh := θ = 2π

k

n: −n

4≤ k <

n

4; k ∈ Z ⊂ [−π

2,π

2),

Θhighh := Θh \Θlow

h ⊂ [−π, π) \ [−π

2,π

2). (3.38)

The LFA formalism is based on the grid functions (3.36). However, instead of consid-ering only a finite number of these functions (varying θ in the finite set Θh), we willallow θ to vary continuously in [−π, π). This makes the formulation easier and moreconvenient. Correspondingly, low frequencies θ vary in [π/2, π/2) and high frequenciesvary θ in [−π, π) \ [π/2, π/2).

Remark 3.1. For clarification, if all θ vary continuously in [−π, π), the ϕh(θ, x) nolonger fulfill the discrete periodic boundary conditions in [0, 1] (only those ϕh withθ ∈ Θh fulfill them). Consequently, we will consider the ϕ(θ, ·) on the infinite gridGh := x : x = xj = jh; j ∈ Z.

3.2 Local Fourier Analysis

Local Fourier analysis (or local mode analysis) was introduced by Brandt in [6] anddeveloped further in several papers (see [8] and the references given there). Thistheoretical approach can be regarded from different points of view and has severalobjectives. We here consider it as a general tool for providing quantitative and real-istic insight into the smoothing properties of relaxation methods and the convergenceproperties of two-grid methods. Our notation is related to the notation and philosophyfrom [21, 22, 25].

3.2.1 Formal Tools of Local Fourier Analysis

In describing the fundamental ideas of LFA, we confine ourselves to the 1D case andto standard coarsening (H = 2h). But it can be immediately carried to d dimensions.The local Fourier analysis is based on certain idealized assumptions and simplifications:the boundary conditions are neglected and the problem is considered on regular infinitegrids

Gh = x : x = xj = jh; j ∈ Z. (3.39)

On Gh, we consider a discrete operator Lh corresponding to a difference stencil

Lh = [lκ]h (κ ∈ Z) (3.40)

i.e. Lhuh(x) =∑κ∈J

lκuh(x + κh) (3.41)

3.2. LOCAL FOURIER ANALYSIS 23

with constant coefficients lκ ∈ R (i.e., lκ does not depend on x), which, of course, willusually depend on h. Here, J ⊂ Z is a certain finite index set defining the so-calleddifference stencil.

The fundamental quantities in the LFA are the grid functions

ϕ(θ, x) = eiθx/h for x ∈ Gh. (3.42)

We assume that θ varies continuously in R. We recognize that

ϕ(θ, x) = ϕ(θ′, x) for x ∈ Gh

if and only ifθ = θ′ (mod 2π).

Therefore, it is sufficient to consider

ϕ(θ, x) with θ ∈ [−π, π).

The grid functions ϕ for −π ≤ θ < π are linearly independent on Gh. Obviously,they are all formal eigenfunctions of Lh:

Lhϕ(θ, x) = Lh(θ)ϕ(θ, x) (x ∈ Gh, θ ∈ [−π, π))

with the corresponding eigenvalues or symbols of Lh

Lh(θ) =∑κ∈J

lκeiθ·k. (3.43)

In addition to Gh, we assume an infinite coarse grid

GH = x : x = xj = jH; j ∈ Z (3.44)

is obtained by standard coarsening of Gh (H = 2h).For the smoothing and two-grid analysis, we again have to distinguish high and low

frequency components on Gh with respect to GH . The definition is based on the factthat only those frequency components

ϕ(θ, ·) with − π

2≤ θ <

π

2

are distinguishable on GH . For each θ′ ∈ [−π/2, π/2), the other frequency compo-nent ϕ(θ, ·) with θ ∈ [−π, π) \ [−π/2, π/2) coincide on GH with ϕ(θ′, ·) and are notdistinguishable (not visible) on GH . Actually, we have

ϕ(θ, x) = ϕ(θ′, x) for x ∈ GH if and only if θ = θ′ (mod π). (3.45)

This leads to the following definition.

Definition 3.2. (high and low frequencies for standard coarsening)

ϕ low frequency component ⇐⇒ θ ∈ Θlow := [−π

2,π

2)

ϕ high frequency component ⇐⇒ θ ∈ Θhigh := [−π, π) \ [−π

2,π

2) (3.46)

This definition is analogous to (2.13) and (3.38), respectively. In addition to speak-ing of high and low frequency components ϕ(θ, ·), we will, for simplicity, sometimescall the corresponding θs high frequency or low frequency.


3.2.2 Smoothing Analysis

According to Definition 3.2, we distinguish high and low frequency components ϕ(θ, ·)and define the smoothing factor µloc(Sh) by

Definition 3.3.

µloc = µloc(Sh) := sup|Sh(θ)| : θ ∈ Θhigh. (3.47)

To calculate the smoothing factor for our model problem, we have to compute thesymbol of the smoothing operator

Shϕ(θ, x) = Shϕ(θ, x) (−π ≤ θ < π). (3.48)

Inserting (2.6), we obtain

Shϕ(θ, x) = (I − ωh

2Kh)ϕ(θ, x) = (I − ωh

2Kh)ϕ(θ, x).

Thus,

Sh = I − ωh

2Kh (3.49)

where I and Kh are the symbols of I and Kh, respectively. Rewriting I and Kh instencil forms

I = [0, 1, 0]h, Kh =1

h[−1, 2,−1]h. (3.50)

From (3.43), we get

I = 1, Kh =1

h(2− 2 cos θ). (3.51)

According to (3.49) and (3.51), the symbol Sh(θ) is

Sh(θ) = 1− ω(1− cos θ). (3.52)

For the smoothing factor, we obtain

µloc(Sh(ω)) = max|1− ω|, |1− 2ω|. (3.53)

This is the same result that we obtained in Section 2.1.3 for µ∗(ω). So, ω-Jacobi forsymmetric operators can be treated by both rigorous Fourier analysis and LFA. Theresults are the same, apart from the following slight formal difference: in comparisonwith (2.15), we recognize that the h-dependence of µ(Sh(ω)) has disappeared here.This is due to the fact that θ varies continuously in Θhigh.


3.2.3 Two-grid Analysis

Let us now apply LFA to the two-grid operator (3.6)

MTGMh = Sν2

h MCGCh Sν1

h with MCGCh = I − Ih

HK−1H IH

h Kh.

In order to calculate convergence factors of MTGMh , we will analyze how the operators

Kh, IHh , KH , Ih

H and Sh act on the Fourier components ϕ(θ, ·). The analysis uses thefact that doubles of the ϕ(θ, ·) coincide on GH . For any low frequency θ ∈ Θlow =[−π/2, π/2), we consider the frequencies

θ0, θ1

where

θ1 :=

θ0 + π if θ0 < 0θ0 − π if θ0 > 0

(3.54)

As mentioned in (3.45), for any low frequency the above defined frequencies coincideon the coarse grid. Interpreting the Fourier components as coarse-grid functions gives

ϕh(θ0, x) = ϕ2h(2θ

0, x) = ϕ2h(2θ1, x) with θ0 ∈ Θlow, x ∈ G2h. (3.55)

Definition 3.4. (harmonics for standard coarsening)The corresponding two ϕ(θα, ·) (and sometimes also the corresponding frequencies)θα are called harmonics (of each other). For a given θ = θ0 ∈ Θlow, we define itstwo-dimensional space of harmonics by

Eθh := span[ϕ(θα, ·) : α ∈ 0, 1]. (3.56)

It will turn out below that the two-grid operator MTGMh leaves the spaces Eθ

h invari-ant yielding a simple block-diagonal representation for MTGM

h consisting of (2 × 2)-blocks

MTGMh |Eθ

h=: MTGM

h (θ) (θ ∈ Θlow) (3.57)

where MTGMh (θ) denotes the representation of the two-grid operator on Eθ

h. Theasymptotic convergence factor ρloc is then defined as follows:

ρloc(MTGMh ) = supρ(MTGM

h ) : θ ∈ Θlow, (3.58)

where ρloc(MTGMh ) is the spectral radius of the (2× 2)-matrix MTGM

h .

We will now derive the representation of MTGMh by (2× 2)-matrices by calculating

the Fourier representation of each multigrid component with respect to the harmonicsEθ

h separately. For clarity, we now write 2h instead of H.

• Fourier representations of operators Kh and IThe treatment of the fine-grid discretization Kh is particularly simple, as theFourier components are eigenfunctions for constant coefficient operators Kh=[kκ]h.


From (3.43) we immediately obtain the Fourier representation of Kh with respectto the harmonics Eθ

h:

Kh(θ) =

(Kh(θ

0) 0

0 Kh(θ1)

)(3.59)

with Fourier symbols Kh(θα) = 1/h(2− 2 cos θα).

Trivially, the identity operator I is represented by the (2× 2)-identity matrix onEθ

h.

• Fourier representation of restriction operator I2hh

The transfer operator I2hh is characterized by a stencil

I2hh =[rκ]

2hh , i.e. I2h

h uh(x) =∑κ∈J

rκuh(x + κh), (x ∈ G2h). (3.60)

We have x + κh ∈ Gh for x ∈ G2h. For the Fourier components this leads to

(I2hh ϕh(θ

α, ·))(x) =∑κ∈J

rκϕh(θα, x + κh) =

∑κ∈J

rκei(x+κh)θα/h

=∑κ∈J

rκeiκθα

ϕh(θα, x)

=∑κ∈J

rκeiκθα

ϕ2h(2θ0, x) (x ∈ G2h) (3.61)

with Fourier symbol

I2hh (θα) :=

∑κ∈J

rκeiκθα

(3.62)

The Fourier representation of I2hh is given by a (1× 2)-matrix

I2hh (θ) = [I2h

h (θ0), I2hh (θ1)]. (3.63)

For the model problem, the Fourier symbol of I2hh = [1/2, 1, 1/2]2h

h is

I2hh (θα) = 1 + cos θα. (3.64)

• Fourier representation of prolongation operator Ih2h

The linear interpolation defined in (2.24) can be expressed in stencil notation

Ih2h =]pκ[

h2h=]1/2, 1, 1/2[h2h (3.65)

The stencil elements correspond to weights in a distribution process. Therefore,the brackets are reversed . For example, each coarse-grid value v2h(x) (x ∈ G2h)is distributed to its two neighbors weighted by p−1 = p1 = 1/2 to obtain fine-gridvalues for (Ih

2hv2h(x))(x) with x ∈ Gh \G2h.


We now investigate the mapping of coarse-grid Fourier components ϕ2h(2θ0, ·)

onto the fine grid by linear interpolation Ih2h.

Assuming x ∈ G2h leads to

(Ih2hϕ2h(2θ

0, ·))(x) = ϕ2h(2θ0, x) = ϕh(θ

0, x). (3.66)

Using ϕ2h(2θ0, x) = ϕh(θ

0, x) (x ∈ G2h, θ0 ∈ Θlow) we obtain for x ∈ Gh \G2h

(Ih2hϕ2h(2θ

0, ·))(x) =1

2(ϕh(θ

0, x− h) + ϕh(θ0, x + h))

=1

2(e−iθ0

+ eiθ0

)ϕh(θ0, x)

= cos θ0ϕh(θ0, x). (3.67)

Then we obtain the following mapping of ϕ2h(2θ0, ·) onto the fine grid:

(Ih2hϕ2h(2θ

0, ·))(x) :=

ϕh(θ

0, x) for x ∈ G2h

cos θ0ϕh(θ0, x) for x ∈ Gh \G2h

(3.68)

For the harmonics, it can be easily established that

ϕh(θ0, x) :=

ϕh(θ

1, x) for x ∈ G2h

−ϕh(θ1, x) for x ∈ Gh \G2h

(3.69)

Furthermore, we introduce two functions ψ1(x) and ψ2(x) defined by:

ψ1(x) =1

2(ϕh(θ

0, x) + ϕh(θ1, x)),

ψ2(x) =1

2(ϕh(θ

0, x)− ϕh(θ1, x)). (3.70)

The following relations are valid:

ψ1(x) =

ϕh(θ

0, x) x ∈ G2h

0 x ∈ Gh \G2h,ψ2(x) =

0 x ∈ G2h

ϕh(θ0, x) x ∈ Gh \G2h

(3.71)Due to (3.71), (3.68) can be rewritten for an arbitrary x ∈ Gh as a linear com-bination of ϕh(θ

α, x):

(Ih2hϕ2h(2θ

0, ·))(x)

= ψ1(x) + cos θ0ψ2(x)

=1

2(1 + cos θ0)ϕh(θ

0, x) +1

2(1− cos θ0)ϕh(θ

1, x). (3.72)

The symbol for linear interpolation is

Ih2h(θ

α) =1

2

∑κ∈J

pκeiθα·κ. (3.73)

The Fourier representation of Ih2h is then given by a (2× 1)-matrix:

Ih2h(θ

0) =

[Ih2h(θ

0)

Ih2h(θ

1)

]=

1

2

[1 + cos θ0

1− cos θ0

]. (3.74)


• Fourier representation of coarse-grid discretization K2h

The stencil notation of K2h is given by

K2h =1

2h[−1, 2,−1]2h (3.75)

For θ0 ∈ Θlow, we thus have

K2hϕ2h(2θ0, ·) = K2h(2θ)ϕ2h(2θ

0, ·) (3.76)

with the symbol

K2h(2θ) =∑κ∈J

kκ,2hei2θ·k =

1

2h(2− 2 cos(2θ0)). (3.77)

• Fourier representation of smoothing operator Sh

From Section 3.2.2 we know all Fourier components ϕh(θ0, ·) are formal eigen-

functions of Sh, and Sh can be represented by the diagonal (2× 2)-matrix

Sh(θ) =

(Sh(θ

0) 0

0 Sh(θ1)

)(3.78)

with Sh(θα) = 1− ω(1− cos θα).

Inserting the representations of I, Ih2h, K2h, I2h

h and Kh and using the fact cos θ =cos θ0 = − cos θ1 (θ = θ0 ∈ Θlow) we obtain the representation of coarse grid correctionoperator MCGC

h on Eθh

MCGCh (θ) = I − Ih

2h(θ)(K2h(θ))−1I2h

h (θ)Kh(θ)

=

[1 00 1

]−

[cos2 θ

2cos2 θ

2

sin2 θ2

sin2 θ2

]

=

[sin2 θ

2cos2 θ

2

sin2 θ2

cos2 θ2

]. (3.79)

From the representation of MTGMh

MTGMh (θ) = Sh(θ)

ν2MCGCh (θ)Sh(θ)

ν1 (3.80)

we obtain the asymptotic convergence factor

ρloc(ν; ω) = supρν(ξ) : 0 < ξ ≤ 1

2 (3.81)

with

ρν(ξ) = |ξ(1− 2ωξ)ν + (1− ξ)(1− 2ω + 2ωξ)ν |, ξ = sin2 θ

2. (3.82)

This is the same result that we obtained in Section 3.1.1 for ρ∗(ν; ω) (see (3.30))derived by rigorous Fourier analysis. In comparison with (3.30), we recognize that theh-dependence of ρ(h, ν; ω) has disappeared here. This is due to the fact that θ variescontinuously.

Chapter 4

Application to an Optimal ControlProblem

In this chapter we study the smoothing factor and two-grid convergence properties foran optimal control problem. First we give a short introduction of the optimal controlproblem which we are going to consider. Then we apply both Fourier analysis andlocal Fourier analysis to the smoothing step and two-grid iteration. The smoother wepresent here is proposed by W.Zulehner (private communication).

Let Ω = (0, 1). Let L2(Ω) and H1(Ω) denote the usual Lebesgue space and Sobolevspace, respectively. In this chapter we consider the following elliptic optimal controlproblem: Find the state y ∈ H1(Ω) and the control u ∈ L2(Ω) such that

J(y, u) = min(z,v)∈H1(Ω)×L2(Ω)

J(z, v)

subject to the state equations

−z′′ = v in Ω ,

z = 0 at Γ,

with cost functional

J(z, v) =1

2‖ z − yd ‖2

L2(Ω) +γ

2‖ v ‖2

L2(Ω)

where yd ∈ L2(Ω) is the desired state and γ > 0 is the weight of the cost of the control(or simply a regularization parameter).

By introducing the adjoint state p ∈ H1(Ω) we get the following optimality system,see e.g.,[24]:1. The adjoint state equation:

− p′′ = −(y − yd) in Ω,

p = 0 at Γ. (4.1)

29

30 CHAPTER 4. APPLICATION TO AN OPTIMAL CONTROL PROBLEM

2. The control equation:γu− p = 0 in Ω. (4.2)

3. The state equation:

− y′′ = u in Ω,

y = 0 at Γ. (4.3)

First we use the control equation (4.2) to eliminate the control u. The reducedoptimality system consists of two PDEs, the adjoint state equation and state equation,where u is replaced by γ−1p. The weak formulation of this problem leads to thevariational problem: Find (y, p) ∈ Y ×Q = H1

0 (Ω)×H10 (Ω) such that

(y, w)L2(Ω) + (p, w)H1(Ω) = 〈yd, w〉 for all w ∈ Y,

(y, q)H1(Ω) − γ−1(p, q)L2(Ω) = 0 for all q ∈ Q.

Let Th be a subdivision of the interval Ω = (0, 1) into subintervals with grid sizeh = 1/n as in Chapter 1. We consider the following discretization by continuous andpiecewise linear finite elements:

Yh = w ∈ C(Ω) : w|T ∈ P1 for all T ∈ Th, w(0) = w(1) = 0 ⊂ Y = H10 (Ω),

Qh = q ∈ C(Ω) : q|T ∈ P1 for all T ∈ Th, q(0) = q(1) = 0 ⊂ Q = H10 (Ω),

where P1 denotes the polynomials of total degree less or equal to 1. Then we obtainthe following discrete variational problem: Find yh ∈ Yh and ph ∈ Qh such that

(yh, w)L2(Ω) + (ph, w)H1(Ω) = 〈yd, w〉 for all w ∈ Y,

(yh, q)H1(Ω) − γ−1(ph, q)L2(Ω) = 0 for all q ∈ Q. (4.4)

Finally, by introducing the standard nodal basis, we obtain the following saddlepoint problem in matrix-vector notation:

Ah

(y

h

ph

)=

(f

h

0

)with Ah =

(Mh Kh

Kh − 1γMh

), (4.5)

where Kh denotes the stiffness matrix representing the H1(Ω) scalar product on Yh.It is the same as in (2.3). Mh denotes the mass matrix representing the L2(Ω) scalarproduct on Yh:

Mh =h

6

4 11 4 1

1. . . . . .. . . . . . 1

1 4

(n−1)×(n−1)

(4.6)

4.1. FOURIER ANALYSIS FOR SMOOTHING OPERATOR 31

4.1 Fourier analysis for Smoothing Operator

4.1.1 Smoothing Operator

We analyze the following preconditioned Richardson method for the smoothing step:

(y

(j+1)h

p(j+1)h

)=

(y

(j)h

p(j)h

)+ τA−1

h

[(fh

0

)−Ah

(y

(j)h

p(j)h

)]τ ∈ (0, 2) (4.7)

where the preconditioner Ah is given by

Ah =

(Mh Kh

Kh − 1γMh

)(4.8)

with Kh = α diag(Kh). The scaling parameter α is chosen small enough to ensure

Kh ≥ Kh. (4.9)

In our example, we fix α = 2. Then

Kh = βI =4

hI

fulfills the assumption (4.9) since λmax(Kh) ≤ 4/h.The corresponding smoothing operator is

SSSh = I − τA−1h Ah. (4.10)

This smoother was proposed by W.Zulehner (private communication).

4.1.2 Generalized Eigenvalue Problem

In order to analyze the smoothing properties, we need to know the eigenvalues λ(SSSh)of the smoothing operator SSSh. Since

λ(SSSh) = 1− τλ(A−1h Ah), (4.11)

we have to study the generalized eigenvalue problem

Ahx = χAhx. (4.12)

This is equivalent to(Ah − χAh)x = 0. (4.13)

Since the two matrices Mh and Kh share the same eigenfunctions

ϕkh(Mh) = ϕk

h(Kh) = (√

2h sin(kjπh))n−1j=1 k = 1, ..., n− 1, (4.14)


we denote

P = [ϕ1h, ϕ

2h, . . . , ϕ

n−2h , ϕn−1

h ]. (4.15)

So both Mh and Kh can be diagonalized through similarity transformations:

P−1MhP = diagλk, P−1KhP = diagµk k = 1, ..., n− 1, (4.16)

where

λk = λk(Mh) =h

3(2 + cos(kπh)),

µk = λk(Kh) =4

hsin2(

1

2kπh). (4.17)

Then, we apply the following similarity transformation to the matrix Ah−χAh anduse (4.16)

[P−1 00 P−1

](Ah − χAh)

[P 00 P

]

=

[P−1 00 P−1

] [(1− χ)Mh Kh − χβIKh − χβI − 1

γ(1− χ)Mh

] [P 00 P

]

=

[(1− χ)diagλk diagµk − χβIdiagµk − χβI − 1

γ(1− χ)diagλk

]. (4.18)

We denote

a = (1− χ)diagλk, b = diagµk − χβI, (4.19)

then the generalized eigenvalue problem (4.12) reduces to the solution of

det

([a bb − 1

γa

])= 0. (4.20)

It can be easily proved that χ = 1 is not an eigenvalue of (4.18), which means a isnonsingular, then from (4.20) we get

det

([a b0 − 1

γa− ba−1b

])= 0,

which is equivalent to

det(−1

γa− ba−1b) = 0. (4.21)

Inserting (4.19) and (4.17) into (4.21) we get the equation

− 1

γ(1− χk)

2λ2k + (µk − βχk)

2 = 0. (4.22)

4.1. FOURIER ANALYSIS FOR SMOOTHING OPERATOR 33

Solving this equation we finally obtain the eigenvalues of A−1h Ah:

χk =(λ2

k + γβµk)± i[√

γβλk −√γµkλk

]

λ2k + γβ2

. (4.23)

Then, the eigenvalues of SSSh are given by the relation

λk(S) = 1− τχk =

(1− τ(λ2

k + γβµk)

λ2k + γβ2

)± i

τ(√

γβλk −√γµkλk)

λ2k + γβ2

, (4.24)

with β = 4/h, λk and µk as in (4.17).

4.1.3 Smoothing Factor

Theorem 4.1. The asymptotic smoothing factor for the smoother given in (4.10) is

µ∗(τ) = max|1− τ |, |1− τ

2|

(4.25)

The choice τopt = 4/3 is optimal in the following sense:

minτ∈(0,2)

µ∗(τ) = µ∗(4

3) =

1

3. (4.26)

Proof : From (4.24) we obtain

|λk(S)|2 = 1− (2τ − τ 2)λ2k + γ(2τβµk − τ 2µ2

k)

λ2k + γβ2

. (4.27)

We assume

|λk(S)|2 ≤ 1− δ, ∀γ ∈ (0,∞), k =n

2, . . . , n− 1, h ∈ (0,

1

2) (4.28)

where δ 6= δ(k, h). We will compute δ in the following. For all h ∈ (0, 1/2], k ∈n/2, . . . , n− 1 and γ ∈ (0,∞), we have

|λk(S)|2 ≤ 1− δ ⇔ 1− (2τ − τ 2)λ2k + γ(2τβµk − τ 2µ2

k)

λ2k + γβ2

≤ 1− δ

⇔ (2τ − τ 2)λ2k + γ(2τβµk − τ 2µ2

k) ≥ (λ2k + γβ2)δ

⇔ λ2k(2τ − τ 2 − δ) + γ(2τβµk − τ 2µ2

k − β2δ) ≥ 0

LetI1 = λ2

k(2τ − τ 2 − δ), I2 = (2τβµk − τ 2µ2k − β2δ),

thus|λk(S)|2 ≤ 1− δ ⇔ I1 + γ I2 ≥ 0, ∀γ, h, k. (4.29)


Since γ ∈ (0,∞), the relation (4.29) is equivalent to I1 ≥ 0 and I2 ≥ 0. (To see this,we first let γ → 0 and obtain I1 ≥ 0. Then rewrite (4.29) as 1

γI1 + I2 ≥ 0 and let

γ →∞ we obtain I2 ≥ 0.) In the following, we consider I1 ≥ 0 and I2 ≥ 0 separately.

I1 ≥ 0 ⇔ λ2k(2τ − τ 2 − δ) ≥ 0

⇔ δ ≤ 2τ − τ 2 (4.30)

andI2 ≥ 0 ⇔ 2τβµk − τ 2µ2

k − β2δ ≥ 0 ∀h, k (4.31)

Inserting β = 4h, µk = 4

hsin2(1

2kπh) = 2

h(1− cos kπh) into (4.31), we obtain

I2 ≥ 0 ⇔ 16

hτ(1− cos kπh)− 4

h2τ 2(1− cos kπh)2 − 16

h2δ ≥ 0

⇔ −τ 2

4(1− cos kπh)2 + τ(1− cos kπh)− δ ≥ 0 ∀h, k. (4.32)

We denote

f(y) = −τ 2

4y2 + τ y − δ with y ∈ [1, 2), and yk = 1− cos kπh. (4.33)

ThenI2 ≥ 0 ⇔ inf

h,kf(yk) ≥ 0 (4.34)

Since for k = n/2, . . . , n− 1,

1 = y 12≤ · · · ≤ yn−1 = 1 + cos πh

and f(y) is convex, we have

infh,k

f(yk) = infh

mink

f(yk) = infh

minf(yn

2), f(yn−1)

= infh

min−τ 2

4+ τ − δ,−τ 2

4(1 + cos πh)2 + τ (1 + cos πh)− δ

= min

−τ 2

4+ τ − δ, 2τ − τ 2 − δ

(4.35)

Hence

I1 + γ I2 ≥ 0 ⇔ δ ≤ min

2τ − τ 2,−τ 2

4+ τ

. (4.36)

Inserting (4.36) into (4.28) and take supremum over h, we get

sup0≤h≤1/2

mink∈n/2,...n−1

|λk(S)|2 = max

(1− τ)2, (1− τ

2)2

. (4.37)

Finally, according to the definition of asymptotic smoothing factor in (2.14), we obtain

µ∗(τ) = max|1− τ |, |1− τ

2|

.

¤

4.2. FOURIER TWO-GRID ANALYSIS 35

Remark 4.2. The smoothing factor is independent of the regularization parameter γ.This is a strong indication that the multigrid method is additionally robust with respectto the regularization parameter γ. Nevertheless, the theory presented here is not ableto show this robustness.

4.2 Fourier Two-grid Analysis

In Section 3.1.1, we know the matrix Qh (see (3.1)) is unitary and multiplication byQh or Q−1

h does not change the spectral radius. So we transform the matrix MTGMh

by the similarity transformation Q−1h MTGM

h Qh and this matrix is much simpler thanthe original one. Here, we use the same trick. However, instead of multiplication byQ−1

h and Qh we use the block diagonal matrices

(Q−1

h 00 Q−1

h

),

(Qh 00 Qh

). (4.38)

We have

ρ(MMMTGMh ) = ρ(MMMTGM

h ) (4.39)

for

MMMTGMh =

(Q−1

h 00 Q−1

h

)MMMTGM

h

(Qh 00 Qh

). (4.40)

From

MMMTGMh = SSSh

ν2MMMCGCh SSSh

ν1 , (4.41)

we obtain

MMMTGMh = SSSh

ν2MMMCGCh SSSh

ν1 . (4.42)

We separately consider the coarse grid operator MMMCGCh and the smoothing operator SSSh.

1. Coarse grid correction MMMCGCh :

MMMCGCh = III − IIIh

2hA−12h III2h

h Ah

= III −(

Ih2h 00 Ih

2h

)(1γ

U−12h M2h U−1

2h K2h

U−12h K2h −U−1

2h M2h

)(I2hh 00 I2h

h

)(Mh Kh

Kh − 1γ

Mh

)

= III −(

1γ

Ih2hU

−12h M2hI

2hh Ih

2hU−12h K2hI

2hh

Ih2hU

−12h K2hI

2hh −Ih

2hU−12h M2hI

2hh

)(Mh Kh

Kh − 1γ

Mh

)

= III −(

T11 T12

T21 T22

)(4.43)


where

U2h =1

γM2

2h + K22h, (4.44)

T11 =1

γIh2hU

−12h M2hI

2hh Mh +

1

γIh2hU

−12h K2hI

2hh Kh, (4.45)

T12 =1

γ(Ih

2hU−12h M2hI

2hh Kh − Ih

2hU−12h K2hI

2hh Mh), (4.46)

T21 = −γ T12, (4.47)

T22 = T11, (4.48)

and Ih2h and I2h

h are defined as (2.25) and (2.22).Since

MMMCGCh =

(Q−1

h 00 Q−1

h

)MMMCGC

h

(Qh 00 Qh

)

= III −(

Q−1h T11Qh Q−1

h T12Qh

Q−1h T21Qh Q−1

h T22Qh

), (4.49)

we deal with each component separately. We start with the first term in T11:

Q−1h

1

γIh2hU

−12h M2hI

2hh MhQh

=1

γQ−1

h Ih2hQ2hQ

−12h (U−1

2h M2h)Q2hQ−12h I2h

h QhQ−1h MhQh

= Ih2h(

1

γΛ2

2M + Λ22K)−1Λ2M I2h

h ΛM , (4.50)

where Ih2h and I2h

h are as in(3.22) and (3.19),

Λ2M = diag λ(M2h), Λ2K = diag λ(K2h), (4.51)

and

ΛM = diag λ1(Mh), λn−1(Mh), . . . , λn2−1(Mh), λn

2+1(Mh), λn

2(Mh). (4.52)

The other components are treated in the same way. Thus, we obtain the representa-tion of coarse grid operator MMMCGC

h .

2. Smoothing operator SSSh:

SSSh = I − τA−1h Ah

= I − τ

(1γ

V −1Mh V −1βI

V −1βI −V −1Mh

)(Mh Kh

Kh − 1γ

Mh

)(4.53)

where

Vh =1

γ(M2

h + β2I). (4.54)

4.3. LOCAL FOURIER ANALYSIS FOR SMOOTHING OPERATOR 37

It is similar as we have done in the previous step to get the representation of smoothingoperator SSSh.

As soon as we obtain the representations for MMMCGCh and SSSh, we then can analyze

the properties of two-grid operator according to (4.42).

4.3 Local Fourier Analysis for Smoothing Operator

Local Fourier analysis for systems of q PDEs is similar to the scalar case we analyzedin previous chapter. Here, we are dealing with vector-valued Fourier components

ϕϕϕh(θ, ·) := ϕh(θ, ·) · I I = (1, . . . , 1)T ∈ Rq (4.55)

and the Fourier symbols

LLLh (θ) =

L1,1h (θ) · · · L1,q

h (θ)... · · · ...

Lq,1h (θ) · · · Lq,q

h (θ)

(4.56)

are given by (q × q)-matrices consisting of scalar Fourier symbols.To analyze the smoothing operator by local Fourier analysis, we have to compute

the symbol of the operator SSSh

SSSh = I − τA−1h Ah = I − τ(Ah)

−1Ah.

Since the discrete operators Mh and Kh can be written in stencil form, the Fouriersymbols Mh and Kh can be easily computed from the stencil form and are given by

Mh(θ) =h

6(4 + 2 cos θ),

Kh(θ) =1

h(2− 2 cos θ). (4.57)

The Fourier symbols of the remaining discrete operators are just combinations of thetwo fundamental symbols Mh and Kh. According to (4.56) the symbols of the discreteoperators Ah and Ah are then the following (2× 2)-matrices:

Ah =

(Mh Kh

Kh − 1γMh

), Ah =

(Mh

4hI

4hI − 1

γMh

). (4.58)

Now any symbolic package (such as MAPLE) can be used to compute the symbol ofI − τA−1

h Ah:SSSh(θ) = [s(i, j)]2i,j=1 (4.59)

with

s(1, 1) = s(2, 2) = 1− τ4h4 cos θ + h4 cos2 θ + 72γ − 72γ cos θ

4h4 + 4h4 cos θ + h4 cos2 θ + 144γ,


s(2, 1) = −γ s(1, 2) = −γ6τh2(cos2 θ + 3 cos θ + 2)

4h4 + 4h4 cos θ + h4 cos2 θ + 144γ.

and the eigenvalues of this matrix. In the limit case h → 0 we finally obtain for theeigenvalues of SSSh:

λ1 = λ2 = 1− τ

2(1− cos θ) (4.60)

The smoothing factor µloc is then given by

µloc(τ) = max|1− τ |, |1− τ

2|. (4.61)

In order to obtain the best smoothing properties, the term µloc(τ) has to be minimizedover the admissible range of the relaxation parameter τ :

minτ∈(0,2)

µloc(τ) = µloc(τopt) =1

3, with τopt =

4

3. (4.62)

This is the same result that we obtained in Section 4.1.3 using rigorous Fourier analysis.

4.4 Local Fourier Two-grid Analysis

Local Fourier two-grid analysis for systems of equations is based on the vector-valuedFourier components ϕϕϕh(θ, ·) introduced in Section 4.3 and the corresponding spacesof harmonics, EEEθ

h with θ ∈ Θlow, which are defined analogously to Definition 3.4,replacing the scalar Fourier components by its vector-valued counterparts.

Similar as we treat the model problem, we will derive the representation of MMMTGMh

by (4× 4)-matrices by considering each multigrid component separately,

MMMTGMh (θ) = (SSSh(θ))

ν2MMMCGCh (θ)(SSSh(θ))

ν1 (4.63)

with the representation of coarse-grid correction operator MMMCGCh

MMMCGCh (θ) = III − IIIh

2h(θ)A−12h (θ)III2h

h (θ)Ah(θ). (4.64)

• Fourier representations of operators Ah and III:

Ah(θ) = bdiagAh(θ0),Ah(θ

1)Inserting the symbols Ah from (4.58) we obtain the Fourier representation of Ah:

Ah(θ) =

Mh(θ0) Kh(θ

0) 0 0Kh(θ

0) − 1γMh(θ

0) 0 0

0 0 Mh(θ1) Kh(θ

1)0 0 Kh(θ

1) − 1γMh(θ

1)

(4.65)

with Mh(θα) and Kh(θ

α) given by (4.57).For identity operator, the representation is

III = diag1, . . . , 1 ∈ C4×4. (4.66)

4.4. LOCAL FOURIER TWO-GRID ANALYSIS 39

• Fourier representation of restriction operator III2hh :

III2hh (θ) =

(III2h

h (θ0), III2hh (θ1)

)=

[I2hh (θ0) 0 I2h

h (θ1) 0

0 I2hh (θ0) 0 I2h

h (θ1)

](4.67)

The Fourier symbols of the restriction operator in connection with systems ofequations are given by (2×2)-matrices, similar as for the discretization operators(see (4.56)). They are composed of scalar Fourier symbols

III2hh (θα) =

[I2hh (θα) 0

0 I2hh (θα)

](4.68)

with I2hh (θα) as in (3.62).

• Fourier representation of prolongation operator IIIh2h:

IIIh2h(θ) =

[IIIh

2h(θ0)

IIIh2h(θ

1)

]=

Ih2h(θ

0) 0

0 Ih2h(θ

0)

Ih2h(θ

0) 0

0 Ih2h(θ

1)

. (4.69)

The Fourier symbols are given by the following (2× 2)-matrices

IIIh2h(θ

α) =

[Ih2h(θ

α) 0

0 Ih2h(θ

α)

], (4.70)

where Ih2h(θ

α) is the same as in (3.73).

• Fourier representation of coarse-grid discretization A2h:

A2h(θ) = A2h(2θ) =

[M2h(2θ) K2h(2θ)K2h(2θ) − 1

γM2h(2θ)

], (4.71)

with the symbols of K2h and M2h given by

K2h(2θ) =1

2h(2− 2 cos(2θ0)), and

M2h(2θ) =2h

6(4 + 2 cos(2θ0)). (4.72)

• Fourier representation of smoothing operator SSSh:

SSSh(θ) = bdiagSSSh(θ0),SSSh(θ

1) ∈ C4×4 (4.73)

with Fourier symbol SSSh(θα) given in (4.59).


Inserting the representations of Ah, III, III2hh , IIIh

2h and A2h into (4.64) we immediatelyobtain the Fourier representation of MMMCGC

h .

Since

ρloc((SSSh(θ))ν2MMMCGC

h (θ)(SSSh(θ))ν1) = ρloc(MMM

CGCh (θ)(SSSh(θ))

ν) (ν = ν1 + ν2) (4.74)

we finally arrive at the two-grid convergence factor

ρloc(ν; τ) = ρloc(MMMTGMh ) = supρloc(MMM

TGMh (θ)) : θ ∈ Θlow (4.75)

with Θlow as in (3.46). Here, ρ depends on ν and τ !

Using MAPLE we can compute the coefficients of the corresponding blocks MMMTGMh

and the eigenvalues. In the following, we give the two-grid convergence factor in thelimit case h → 0 and optimal choices of parameter τ for special cases ν = 1 and ν = 2.


ρloc(1; τ) = sup| − 1

2τ cos2(θ)− 1

2τ + 1| : θ ∈ [−π

2,π

2)

= max|1− τ |, |1− τ

2|

The choice τ = 4/3 is optimal in the following sense:

infρloc(1; τ) : 0 ≤ τ ≤ 2 = ρloc(1;4

3) =

1

3.


ρloc(2; τ) = sup|(34τ 2 − τ) cos2(θ) +

1

4τ 2 − τ + 1| : θ ∈ [−π

2,π

2)

= max(1− τ)2, (1− τ

2)2

The choice τ = 4/3 is optimal in the following sense:

infρloc(2; τ) : 0 ≤ τ ≤ 2 = ρloc(2;4

3) =

1

9.

The optimal choices of τ in these two cases are the same with the choice in thesmoothing analysis in (4.62). This is a strong indication that τ = 4/3 is a good choicein the multigrid method. However, we are not able to predict the optimal choice of τfor the multigrid method.

4.5. NUMERICAL RESULTS 41

4.5 Numerical Results

Next we present some numerical results for the domain Ω = (0, 1) and homogeneousdesired state yd = 0. The coarsest grid consists of only one inner point, i.e., withgrid size h0 = 1/2. For the first series of experiments the regularization parameter γ

was set equal to 1. The starting values for y(0)k and p

(0)k for the exact solution yk = 0

and pk = 0 were randomly chosen. The discretized problem was solved by a multigriditeration with a V-cycle and m pre- and m post-smoothing steps. The multigrid iter-ation was performed until the Euclidean norm of the residual was reduced by a factorε = 10−8.

Table 4.1 contains the number of unknowns nk (total number of unknowns is nk +nk), the number of iterations it and the convergence rates q depending on the level kand the number m of smoothing steps.

smoothing stepsk nk 0+1 1+0 1+1 2+2 3+3 5+5

5 63 26 0.49 27 0.49 14 0.26 9 0.11 7 0.074 6 0.0456 127 26 0.49 27 0.49 14 0.26 9 0.11 7 0.074 6 0.0457 255 26 0.50 27 0.50 14 0.26 9 0.11 7 0.074 6 0.0458 511 26 0.50 27 0.50 14 0.26 9 0.11 7 0.074 6 0.0459 1023 26 0.50 27 0.50 14 0.26 9 0.11 7 0.074 6 0.045

Table 4.1: V-cycle convergence rates with τ = 1

In Section 4.1.3 we discussed the choice of the relaxation parameter τ . Table 4.2shows the convergence rates for the overrelaxed smoother with τ = 1.33. However, theconvergence rates are much better than the rates for the original smoothing iterationwith τ = 1. The optimal choice of τ for smoothing operator and two-grid iteration isa good choice for multigrid method.


5 63 16 0.32 17 0.32 11 0.18 7 0.083 6 0.054 5 0.0336 127 16 0.32 17 0.31 11 0.18 7 0.083 6 0.054 5 0.0337 255 17 0.33 17 0.32 11 0.18 7 0.083 6 0.054 5 0.0338 511 17 0.33 17 0.32 11 0.18 7 0.083 6 0.054 5 0.0339 1023 17 0.33 17 0.33 11 0.18 7 0.083 6 0.054 5 0.033

Table 4.2: V-cycle convergence rates with τ = 1.33

All numerical experientments shown so far were performed for the regularizationparameter γ = 1. Table 4.3 shows the results of numerical experiments obtained atgrid level 7 for values of γ ranging from 10 down to 10−5. We can see that the behavior


of convergence rate q with respect to γ is as we have expected and the method proposedis robust in the regularization parameter γ.

smoothing stepsγ 1+1 10+10

10 14 0.26 5 0.0221 14 0.26 5 0.022

10−1 14 0.26 5 0.02210−2 14 0.26 5 0.02410−3 14 0.28 5 0.02910−4 15 0.28 6 0.02810−5 15 0.28 5 0.027

Table 4.3: γ-dependence, τ = 1

Finally, Table 4.4 and Table 4.5 show the convergence rates for W-cycle iterationusing τ = 1 and τ = 1.33, respectively.


5 63 26 0.495 26 0.495 13 0.245 8 0.08 6 0.054 6 0.0336 127 26 0.495 26 0.495 13 0.245 8 0.08 6 0.054 6 0.0337 255 26 0.495 26 0.495 13 0.245 8 0.08 6 0.054 6 0.0338 511 26 0.495 26 0.495 13 0.245 8 0.08 6 0.054 6 0.0339 1023 26 0.495 26 0.495 13 0.245 8 0.08 6 0.054 6 0.033

Table 4.4: W-cycle convergence rates with τ = 1


5 63 16 0.328 16 0.328 9 0.113 7 0.059 6 0.041 5 0.0256 127 16 0.328 16 0.328 9 0.113 7 0.059 6 0.041 5 0.0257 255 16 0.328 16 0.328 9 0.113 7 0.059 6 0.041 5 0.0258 511 16 0.328 16 0.328 9 0.113 7 0.059 6 0.041 5 0.0259 1023 16 0.328 16 0.328 9 0.113 7 0.059 6 0.041 5 0.025

Table 4.5: W-cycle convergence rates with τ = 1.33

Chapter 5

Conclusion

In this thesis we first study the smoothing factor and two-grid convergence factor of1D poisson problem using damped Jacobi iteration as smoother. Fourier two-gridanalysis for this model problem was described in [18], but for fixed relaxation param-eter ω = 1/2. Here we generalize the Fourier two-grid analysis for all ω ∈ [0, 1] andstudy the optimal choices of ω when the total smoothing steps are equal to 1 or 2. Incomparison, local Fourier smoothing and two-grid analysis is also presented for modelproblem. And we get the same results for both smoothing factor and two-grid con-vergence factor in the asymptotic sense. Also, from the theoretical point of view theconnections and differences of Fourier analysis and LFA are illustrated.

The main part of this work is application to an optimal control problem where aregularization parameter γ is involved. It would be desirable to achieve robustnessof the smoother proposed by W.Zulehner with respect to this regularization param-eter. Finite element discretization results in two-by-two block system which makesthe convergence analysis more complicated than the scalar 1D model problem. Weobtain the asymptotic smoothing factor and observe its independence with respect tothe regularization parameter γ. However, we cannot conclude from the robustness ofthe smoothing factor that the two or the multigrid methods are robust. Local Fouriertwo-grid analysis is derived and two-grid convergence factor for special cases are ob-tained. Since the Fourier two-grid analysis is very technical, we describe the road mapto derive the two-grid convergence rates by Fourier analysis. Numerical results hereshow the robustness.

The optimal control problem analyzed in Chapter 4 is quite similar to the mixedformulation of the biharmonic equation. Indeed, for the first biharmonic boundaryvalue problem in 1D (beam equation)

− u(4) = f in Ω, u = u′ = 0 at Γ, (5.1)

we obtain the following mixed variational problem by introducing the new unknown

43

44 CHAPTER 5. CONCLUSION

function w = u′′: Find (w, u) ∈ V ×W = H1(Ω)×H10 (Ω) such that

a(w, v) + b(u, v) = 0 for all v ∈ V,

b(w, q) = 〈G, q〉 for all q ∈ W, (5.2)

with

a(w, v) =

∫ 1

0

w(x)v(x)dx,

b(u, v) =

∫ 1

0

u′(x)v′(x)dx,

〈G, q〉 = −∫ 1

0

f(x)q(x)dx.

We again subdivide the interval Ω = (0, 1) into n parts with grid size h = 1n

anddiscrete by continuous and piecewise linear finite elements:

Vh = v ∈ C(Ω) : v|T ∈ P1 for all T ∈ Th,Wh = q ∈ C(Ω) : q|T ∈ P1 for all T ∈ Th, q(0) = q(1) = 0.

So we obtain the following discrete variational problem: Find wh ∈ Vh and uh ∈ Wh

such that

a(wh, v) + b(uh, v) = 0 for all v ∈ Vh,

b(wh, q) = 〈G, q〉 for all q ∈ Wh.

By introducing the same standard nodal basis for all unknowns, we finally obtainthe following saddle point problem in matrix-vector notation:

Ah

(wh

uh

)=

(0g

h

)with Ah =

(Mh KT

h

Kh 0

)(5.3)

where Mh is the (n + 1)× (n + 1) mass matrix

Mh =h

6

2 11 4 1

1 4 1

1. . . . . .. . . . . . 1

1 4 11 2

(n+1)×(n+1)

(5.4)

45

and Kh is the (n− 1)× (n + 1) stiffness matrix

Kh =1

h

−1 2 −1−1 2 −1

−1. . . . . .. . . . . . −1

−1 2 −1−1 2 −1

(n−1)×(n+1)

. (5.5)

The main difference of optimal control problem and the biharmonic problem is: thestiffness matrix Kh in optimal control problem is a square matrix whereas in bihar-monic problem is a rectangular matrix. This is due to the difference in the boundaryconditions. Apply LFA to biharmonic problem will give the same results as for op-timal control problem since we neglect the boundary conditions in LFA. This is notthe result we want to obtain. Future work will focus on the Fourier analysis of thisproblem.

46 CHAPTER 5. CONCLUSION

Bibliography

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[2] N. Bakhvalov. On the convergence of a relaxation method with natural constraintson the elliptic operator. Zh. Vychisl. Mat. Mat. Fiz, 6:861–885, 1966.

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[4] A. Brandt. Multi-level adaptive technique/MLAT/ for fast numerical solution toboundary value problems. In International Conference on Numerical Methods inFluid Mechanics, 3 rd, Paris, France, pages 82–89, 1973.

[5] A. Brandt. Multi-level adaptive techniques. IBM IJ Watson Research Center,Yorktown Heights, New York. IBM Research Report RC6026, 1976.

[6] A. Brandt. Multi-level adaptive solutions to boundary-value problems. Math.Comp., 31(138):333–390, 1977.

[7] A. Brandt. Multigrid solvers for non-elliptic and singular-perturbation steady-state problems. The Weizmann Institute of Science, Rehovot, Israel, 1981.

[8] A. Brandt. Guide to multigrid development. Multigrid methods, 960:220–312,1982.

[9] A. Brandt. Rigorous quantitative analysis of multigrid, I: Constant coefficientstwo-level cycle with L2-norm. SIAM Journal on Numerical Analysis, pages 1695–1730, 1994.

[10] W. Briggs, V. Henson, and S. McCormick. A Multigrid tutorial 2nd ed. SIAM,Philadelphia, 2000.

[11] R. Fedorenko. A relaxation method for solving elliptic difference equations. USSRComput. Math. and Math. Phys, 1(5):1092–1096, 1961.

[12] R. Fedorenko. The speed of convergence of one iterative process. USSR Comp.Math. and Math. Physics, 4:227–235, 1964.

47

48 BIBLIOGRAPHY

[13] W. Hackbusch. Ein iteratives Verfahren zur schnellen Auflosung elliptischerRandwertprobleme. Report 76-12, Institute for Applied Mathematics, Universityof Cologne, West Germany, 1976.

[14] W. Hackbusch. On the convergence of a multi-grid iterations applied to finiteelement equations. Report 77-8, Institute for Applied Mathematics, University ofCologne, West Germany, 1977.

[15] W. Hackbusch. Convergence of multi-grid iterations applied to difference equa-tions. Mathematics of Computation, pages 425–440, 1980.

[16] W. Hackbusch. On the convergence of multi-grid iterations. Beitrage zur Numer.Math, 9:213–239, 1981.

[17] W. Hackbusch. Multi-grid convergence theory. In Multigrid Methods: Proceedingsof the Conference Held at Koln-Porz, November 23-27, 1981, page 177. Springer-Verlag, 1982.

[18] W. Hackbusch. Multi-grid methods and applications. Springer, 1985.

[19] W. Hackbusch and U. Trottenberg. Multigrid Methods, Lecture Notes in Math.,Vol. 960, 1982.

[20] V. Korneev. Finite Element Schemes of High Order of Accuracy, Leningrad Uni-versity, Leningrad, 1977.

[21] K. Stuben and U. Trottenberg. Multigrid methods: fundamental algorithms,model problem analysis and applications. In Multigrid methods (Cologne, 1981),volume 960 of Lecture Notes in Math., pages 1–176. Springer, Berlin, 1982.

[22] K. Stueben and U. Trottenberg. On the construction of fast solvers for ellipticequations. In Von Karman Inst. for Fluid Dyn. Computational Fluid Dyn. 161 p(SEE, 83, 1982.

[23] C.-A. Thole and U. Trottenberg. A short note on standard parallel multigridalgorithms for 3d problems. Appl. Math. Comput., 27(2):101–115, 1988.

[24] F. Troltzsch. Optimale Steuerung partieller Differentialgleichungen: Theorie, Ver-fahren und Anwendungen. Vieweg Verlag, 2005.

[25] U. Trottenberg, C. W. Oosterlee, and A. Schuller. Multigrid. Academic PressInc., San Diego, CA, 2001. With contributions by A. Brandt, P. Oswald and K.Stuben.

[26] P. Wesseling. An introduction to multigrid methods. Wiley Chichester, 1992.

[27] R. Wienands and W. Joppich. Practical Fourier analysis for multigrid methods,volume 4 of Numerical Insights. Chapman & Hall/CRC, Boca Raton, FL, 2005.With 1 CD-ROM (Windows and UNIX).

BIBLIOGRAPHY 49

[28] D. Young Jr. Iterative methods for solving partial differential equations of elliptictype (Ph. D. thesis), Harvard University, Cambridge, Mass, 1950.

50 BIBLIOGRAPHY

Curriculum Vitae

Name: Zhou Yao

Nationality: China

Date of Birth: January 7th, 1986

Place of Birth: Hubei Province, China

Education:

2003 - 2007 Bachelor Study, Informatics and Computational MathematicsBeihang University

2007 - 2008 Graduate Study, Applied MathematicsBeihang University Master

2008 - 2009 International Master’s Program for InformaticsAustria, Hagenberg, Johannes Kepler University (JKU)

Special Activities:

November 2006 China Undergraduate Mathematical Contest in ModelingSecond Prize in China

March 2007 America Mathematical Contest in ModelingHonorable Mention.

52 BIBLIOGRAPHY

Eidesstattliche Erklarung

Ich erklare an Eides statt, dass ich die vorliegende Masterarbeit selbstandig und ohnefremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutztbzw. die wortlich oder sinngemaß entnommenen Stellen als solche kenntlich gemachthabe.

Linz, July 2009

————————————————Yao Zhou

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