Analysis of Schwarz methods for discontinuous Galerkin discretizations

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Thesis

Reference

Analysis of Schwarz methods for discontinuous Galerkin discretizations

HAJIAN, Soheil

Abstract

This thesis is conducted in the field of numerical analysis which is part of applied mathematics. More precisely we study some methods to solve certain linear systems. The linear systems that we consider are derived from discretizations of partial differential equations. Such linear systems often inherit the properties of the underlying partial differential equation. For example the corresponding matrix of the linear system is sparse. This property motivates the use of iterative methods for the solution technique of such linear systems, since the multiplication of a sparse matrix with a vector is computationally cheap. In this thesis we propose one such iterative method and prove rigorously its advantage over other iterative methods.

HAJIAN, Soheil. Analysis of Schwarz methods for discontinuous Galerkin discretizations. Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4795

URN : urn:nbn:ch:unige-752254

DOI : 10.13097/archive-ouverte/unige:75225

Available at:

http://archive-ouverte.unige.ch/unige:75225

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GEN ÈVE FACULTÉ DES SCIENCES

Section de math´ematiques Martin J. Gander

Analysis of Schwarz methods for discontinuous Galerkin discretizations

Ph.D. THESIS

presented to the Faculty of Science, University of Geneva for obtaning the degree of Doctor of Mathematics.

by Soheil Hajian

from Tehran

Ph.D. N 4795

Atelier de reproduction de la Section de physique Geneva, June 24, 2015

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ﻪﻣﺪﻘﻣ

يا ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ

رد درﺬ ﮔﯽ ﻣ هﺪ ﻨ ﻧاﻮ ﺧ ﺮ ﻈ ﻧ زا ﻪ ﻣادا رد ﻪ ﮐ يﻪ ﺘ ﺷﺬ ﮔ لﺎ ﺳ رﺎ ﻬ ﭼ ِﯽ ﺸ ﻫوﮋ ﭘ ِرﺎ ﮐ ﻞ ﺻﺎ ﺣ لوا ﺮ ﻈ ﻧ هﺎ ﮕ ﺸ ﻧاد رد ﯽ ﺿﺎ ﯾر يﺮ ﺘ ﮐد يﻮ ﺠ ﺸ ﻧاد ناﻮ ﻨ ﻋ ﻪ ﺑ هﺪ ﻧرﺎ ﮕ ﻧ هدﻮ ﺒ ﻧ ﺮ ﯾﺬ ﭘنﺎ ﮑ ﻣا ناﺮ ﮕ ﯾد ﮏ ﻤ ﮐ نوﺪ ﺑ ﮏ ﺷﯽ ﺑ نآ مﺎ ﻤ ﺗا ﺎ ﻣا ﺪ ﺷﺎ ﺑﯽ ﻣ ﻮ ﻧژ ردﺎ ﻣ و رﺪ ﭘ ﯽ ﻧﺎ ﺒ ﯿ ﺘ ﺸ ﭘ راﺬ ﮕ ﺳﺎ ﭙ ﺳ ﻪ ﻤ ﻫ زا ﺶ ﯿ ﺑ و ﺰ ﯿ ﭼﺮ ﻫ زا ﺶ ﯿ ﭘ .ﺖ ﺳا هﺪ ﻧرﺎ ﮕ ﻧ .ﻢ ﺘ ﺴ ﻫ ﻢ ﻧﺎ ﺑﺮ ﻬ ﻣ گرﺰ ﺑردﺎ ﻣ و ﺰ ﯾﺰ ﻋ ﻮ ﻤ ﻋ و ﻪ ﻤ ﻋ ﮏ ﻤ ﮐ و ﻒ ﻄ ﻟ ،ﯽ ﻨ ﺘ ﺷادﺖ ﺳود رﺪ ﻧﺎ ﮔ ﯽ ﺟ ﻦ ﯿ ﺗرﺎ ﻣدﺎ ﺘ ﺳا يﺎ ﻫ هﺪ ﯾا نوﺪ ﺑ ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﯾا ﺞ ﯾﺎ ﺘ ﻧ دراد نﺎ ﻨ ﯿ ﻤ ﻃا هراﻮ ﻤ ﻫ ﻪ ﮐ دﻮ ﺑ ﻢ ﻫاﻮ ﺧ نﺎ ﺸ ﯾا رادماو ﺖ ﻬ ﺟ نآ زا ﺮ ﺘ ﺸ ﯿ ﺑ ﻦ ﻣ ﯽ ﻟو ﺪ ﺷﯽ ﻤ ﻧ ﯽ ﻧﻮ ﻨ ﮐ ﯽ ﺑﻮ ﺧ ﻪ ﺑ ﺎ ﺑ نﺎ ﺴ ﻤ ﻫ ﺶ ﻫﺎ ﮕ ﻧ رد ﻮ ﺠ ﺸ ﻧاد و دراد ﺖ ﮐرﺎ ﺸ ﻣ ﻮ ﮕ ﺘ ﻔ ﮔ و ﺚ ﺤ ﺑ رد ﯽ ﻧدز لﺎ ﺜ ﻣ يﺮ ﺒ ﺻ ﺎ ﺑ ﺖ ﺳا يدﺮ ﺑرﺎ ﮐ تﺎ ﯿ ﺿﺎ ﯾر زا يا ﻪ ﺧﺎ ﺷﺮ ﯾز ﻪ ﮐ يدﺪ ﻋ ِبﺎ ﺴ ﺣيهزﻮ ﺣ رد ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﯾا .ﺖ ﺳوا دﻮ ﺧ ﯽ ﻄ ﺧ تﻻدﺎ ﻌ ﻣ ﻞ ﺣ ياﺮ ﺑ يدﺪ ﻋ يﺎ ﻫشور ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﯾا رد ﺮ ﺗﻖ ﯿ ﻗد نﺎ ﯿ ﺑ ﻪ ﺑ .ﺖ ﺳا هﺪ ﺷ ﻪ ﺘ ﺷﺎ ﮕ ﻧ تﻻدﺎ ﻌ ﻣ زا ياﻪ ﺘ ﺳد ﻢ ﯿ ﻨ ﮐﯽ ﻣ ﯽ ﺳرﺮ ﺑ ﺎ ﺠ ﻨ ﯾا رد ﺎ ﻣ ﻪ ﮐ ياﯽ ﻄ ﺧ تﻻدﺎ ﻌ ﻣ .دﻮ ﺷﯽ ﻣ ﻪ ﻌ ﻟﺎ ﻄ ﻣ و ﯽ ﺳرﺮ ﺑ ﯽ ﻄ ﺧ تﻻدﺎ ﻌ ﻣ زا ﻪ ﺘ ﺳد ﻦ ﯾا .ﺪ ﻧاهﺪ ﻣآ ﺖ ﺳﺪ ﺑ ﯽ ﺋﺰ ﺟ ﻞ ﯿ ﺴ ﻧاﺮ ﻔ ﯾد تﻻدﺎ ﻌ ﻣ ِﺐ ﯾﺮ ﻘ ﺗ زا ﻪ ﮐ ﺪ ﻨ ﺘ ﺴ ﻫ ﯽ ﻄ ﺧ رﻮ ﻃ ﻪ ﺑ .ﺪ ﻨ ﺘ ﺴ ﻫ ﻪ ﻃﻮ ﺑﺮ ﻣ ﯽ ﺋﺰ ﺟ ﻞ ﯿ ﺴ ﻧاﺮ ﻔ ﯾد يﻪ ﻟدﺎ ﻌ ﻣ ﺎ ﺑ يرﺎ ﯿ ﺴ ﺑ تﺎ ﮐاﺮ ﺘ ﺷا ياراد لﻮ ﻤ ﻌ ﻣ رﻮ ﻃ ﻪ ﺑ داﺪ ﻋا وﺰ ﺟ ﯽ ﮕ ﻤ ﻫ نآ يهﮋ ﯾو ﺮ ﯾدﺎ ﻘ ﻣ و ﺖ ﺳا تﻮ ﻠ ﺧﯽ ﻄ ﺧ ﻢ ﺘ ﺴ ﯿ ﺳ ﻦ ﯾا ﺎ ﺑ ﺮ ﻇﺎ ﻨ ﺘ ﻣ ﺲ ﯾﺮ ﺗﺎ ﻣ ،لﺎ ﺜ ﻣ

ﻞ ﺣ ياﺮ ﺑ ،يدﺪ ﻋ هﺪ ﻧﻮ ﺷ راﺮ ﮑ ﺗ يﺎ ﻫشورسﺎ ﺳا و ﻪ ﯾﺎ ﭘ هﺪ ﻫﺎ ﺸ ﻣ ﻦ ﯾا .ﺪ ﻨ ﺘ ﺴ ﻫ ﺖ ﺒ ﺜ ﻣ و ﯽ ﻘ ﯿ ﻘ ﺣ .ﺖ ﺳا ياﻪ ﻨ ﯾﺰ ﻫ ﻢ ﮐ ِﻞ ﻤ ﻋ رادﺮ ﺑ ﮏ ﯾ رد تﻮ ﻠ ﺧ ﺲ ﯾﺮ ﺗﺎ ﻣ ﮏ ﯾ بﺮ ﺿ اﺮ ﯾز ؛ﺪ ﺷﺎ ﺑﯽ ﻣ ﯽ ﻄ ﺧ تﻻدﺎ ﻌ ﻣ ﻦ ﯿ ﻨ ﭼ

،ﻢ ﯾﻮ ﺷﯽ ﻣ ﺎ ﻨ ﺷآ ﯽ ﻄ ﺧ تﻻدﺎ ﻌ ﻣ ﻪ ﺘ ﺳد ﻦ ﯾا ﻞ ﺣ ياﺮ ﺑ هﺪ ﻧﻮ ﺷراﺮ ﮑ ﺗ شور ﮏ ﯾ ﺎ ﺑ ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﯾا رد .ﻢ ﯿ ﻨ ﮐﯽ ﻣ تﺎ ﺒ ﺛا ﺎ ﻫشور ﺮ ﯾﺎ ﺳ ﻪ ﺑ ﺖ ﺒ ﺴ ﻧ ار نآ ﺖ ﯾﺰ ﻣ ﯽ ﺿﺎ ﯾر يﺎ ﻫراﺰ ﺑا زا هدﺎ ﻔ ﺘ ﺳا ﺎ ﺑ ﺲ ﭙ ﺳ

ﻞ ﺼ ﻓ رد ،هدﺮ ﮐ اﺪ ﯿ ﭘ ﯽ ﯾاﺪ ﺘ ﺑا ﯽ ﯾﺎ ﻨ ﺷآ شور ﻦ ﯾا ﺎ ﺑ هﺪ ﻨ ﻧاﻮ ﺧ ،لﺎ ﺜ ﻣ ندروآ ﺎ ﺑ لوا ﻞ ﺼ ﻓ رد

،نﺎ ﯾﺎ ﭘ رد و دﻮ ﺷﯽ ﻣ هداد حﺮ ﺷ ﯽ ﺋﺰ ﺟ ﻞ ﯿ ﺴ ﻧاﺮ ﻔ ﯾد تﻻدﺎ ﻌ ﻣ ﻞ ﺣ ﯽ ﺒ ﯾﺮ ﻘ ﺗ يﺎ ﻫشور مود ﻪ ﻌ ﻟﺎ ﻄ ﻣ ياﻪ ﻧﺎ ﻔ ﮑ ﺷﻮ ﻣ ِترﻮ ﺼ ﺑ ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﯾا يدﺎ ﻬ ﻨ ﺸ ﯿ ﭘ شور ،مﻮ ﺳ ﻞ ﺼ ﻓ رد

هدﻮ ﺑ ﺪ ﯿ ﻔ ﻣ و ﺮ ﺼ ﺘ ﺨ ﻣ ﻪ ﻣﺎ ﻧنﺎ ﯾﺎ ﭘ ﻦ ﺘ ﻣ ﻪ ﮐ ﺖ ﺳا هﺪ ﯿ ﺷﻮ ﮐ هﺪ ﻧرﺎ ﮕ ﻧ .دﻮ ﺷﯽ ﻣ دﺎ ﺠ ﺳ صﻮ ﺼ ﺨ ﺑ ،ﻢ ﻧﺎ ﺘ ﺳود ﯽ ﻫاﺮ ﻤ ﻫ زا نﺎ ﯾﺎ ﭘ رد .دﺰ ﯿ ﻫﺮ ﭙ ﺑ ﯽ ﯾﻮ ﮔهدﺎ ﯾز زا و

.ﻢ ﯾﺎ ﻤ ﻧﯽ ﻣ ﺮ ﮑ ﺸ ﺗ ،ﻪ ﻣﺪ ﻘ ﻣ ﻦ ﯾا ﺶ ﯾاﺮ ﯾو ﺖ ﺑﺎ ﺑ ﺰ ﯾﺰ ﻋ يرﺎ ﺼ ﻧا تﺬ ﻟ ﺪ ﯾآﯽ ﻣ ﻪ ﻣادا رد ﻪ ﭽ ﻧآ نﺪ ﻧاﻮ ﺧ زا هﺪ ﻨ ﻧاﻮ ﺧ ﻪ ﮐ مراوﺪ ﯿ ﻣا

ﻊ ﻗاو ﺪ ﯿ ﻔ ﻣ ﺶ ﻫوﮋ ﭘ ﻦ ﯾا ﺞ ﯾﺎ ﺘ ﻧ و هدﺮ ﺑ .دﻮﺷ

Martin J. Gander = رﺪﻧﺎﮔ ﯽﺟ ﻦﯿﺗرﺎﻣ numerical analysis = يدﺪﻋ ِبﺎﺴﺣ sparse matrix = تﻮﻠﺧ ﺲﯾﺮﺗﺎﻣ iterative method = هﺪﻧﻮﺷ راﺮﮑﺗ شور

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Preface ¹

The following thesis is the result of my research during the past four years at the University of Geneva. It would not have been possible to achieve this without help of many others. First I should thank my parents and family for their help and support during all these years. Second I should acknowledge that the results of this thesis would not be this much promising without input from my adviser Prof. Martin J. Gander.

I am also grateful to him for his remarkable patience during our discussions and his non-academic guidance.

This thesis is conducted in the field of numerical analysis which is part of applied mathematics. More precisely we study some methods to solve certain linear systems.

The linear systems that we consider are derived from discretizations of partial differential equations. Such linear systems often inherit the properties of the underlying partial differential equation. For example the corresponding matrix of the linear system is sparse. This property motivates the use of iterative methods for the solution technique of such linear systems, since the multiplication of a sparse matrix with a vector is computationally cheap. In this thesis we propose one such iterative method and prove rigorously its advantage over other iterative methods.

The thesis is intended to be short but concise and the author hopes that the reader enjoys reading it. I should also thank my friends for making my PhD experience pleasant: Xavier Morvan, Caroline Giacobino, Máté Juhász, Alexey Talambutsa and Hester Pieters. Many thanks to Caterina Campagnolo for her unlimited kindness and her help with writing the thesis abstract in French. I should also thank my compatriots Parisa Mamooler and Aran Raoufi for amusing conversations in Persian. A special thanks also goes to Justine Simone Louis for bringing sanity to the math department and helping me with French.

1This preface is a loose translation of the Persian preface (previous page).

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Introduction

If you have not yet skipped this introduction¹, let me briefly give an overview of this thesis. For a given partial differential equation (PDE) problem, there exists a vast arsenal of numerical methods to approximate the exact solution of the underlying PDE. These numerical methods usually lead to a large and sparse linear system to be solved. For problems that arise from real applications, the size of the linear system is usually too large to perform adirect solve; for example computing an LU-factorization is too expensive in terms of time and storage. Another approach is to use an iterative method to approximate the solution of the linear system.

In recent decades with the introduction of parallel computers, a huge effort is dedi- cated to design and analyse parallel iterative methods by decomposing the underlying linear system (or equivalently the underlying PDE’s domain) into sub problems and solve them concurrently. These methods are called domain decomposition methods.

Due to the “iterative” nature of domain decomposition methods, the speed of convergence of these methods is of great interest. The objective of this thesis is mainly to design and analyse a faster domain decomposition method for discontinuous Galerkin discretizations and compare it with the existing ones.

1.1 A numerical experiment

The best way to demonstrate a comparison between existing methods and what we will present in this thesis is by example: let us consider the following Poisson problem in two dimensions,

−∆u = f, in Ω = (0, a)×(0, b),

u = 0, on∂Ω, (1.1)

where u is unknown and for the moment f is a given function such that the solution, u, has enough regularity. Therefore we consider an approximation of this unknown by a finite difference (FD) method. To do so, we first partition the domain into cells of size h×h and define a grid function over it, here denoted by u. We denote the total number of grid points by n. As the mesh size, h, tends to zero the number of nodes grows like n = O(h⁻²). For an example see Figure 1.1 with a simple grid and eight unknowns, u = (u₁, u₂, . . . , u₈) and n = 8. The differential operator is approximated

1I usually do skip introductions, but I wrote this chapter keeping in mind that a general audience will read it.

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u₁

u₂ u₃

u₄

u5

u6

u₇

u8

Ω¹ Ω²

Γ

Figure 1.1: An example of a grid with the numbering of unknowns.

byA ∈Rⁿ^×ⁿ:

A:= 1 h²







4 −1 −1

−1 4 −1

−1 4 −1 −1

−1 −1 4 −1

4 −1 −1

−1 4 −1

−1 −1 4 −1

−1 −1 −1 4







, (1.2)

which is a symmetric positive definite (s.p.d.) matrix. This leads to a linear system of the form

Au=f, (1.3)

where f = (f₁, f₂, . . . , f₈) and f_i :=f(x_i, y_i).

In this example the domain is partitioned into two subdomains, Ω1 and Ω2, with the interface Γ. Subdomain one, Ω1, has four unknowns: u1 := (u1, u2, u3, u4), subdomain two, Ω₂, has two unknowns: u₂ := (u₅, u₆). The interface, Γ, contains two unknowns:

u_Γ := (u₇, u₈). In general we can partition (or represent) the matrix A and the linear system (1.3) in the following form



 A₁ A_1Γ A2 A2Γ

A^>_1Γ A^>_2Γ A_Γ







 u₁ u₂ uΓ



=



 f₁ f₂ f_Γ



. (1.4)

As we mentioned earlier, the goal of this thesis is to design and analyse an iterative method to solve the linear system (1.3) or equivalently (1.4). Here we compare two iterative methods. The first one is the so-calledadditive Schwarzmethod (ASM) which is a block Jacobi method applied to (1.4). The additive Schwarz method for this problem is defined as: for a given initial guess, u⁽⁰⁾ ∈Rⁿ, solve for u^(m) until convergence using

Mu^(m) =Nu^(m⁻¹⁾+f, where N =M −A and

M :=



 A₁

A₂ A_2Γ A_2Γ A_Γ



.

The second method is calledoptimized Schwarzmethod (OSM) and is less widely used.

In this thesis we will show that for different discretizations OSM performs better than

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h0 h0/2 h0/4 h0/8 h0/16 h0/32 growth rate

ASM 133 266 531 1055 2111 – h⁻¹

OSM withγ= 1 32 60 112 207 382 699 h⁻¹

OSM withγ= ¹₂(1 +√

h) 9 13 18 24 34 46 h⁻^1/2

Table 1.1: Number of iterations required to reduce the error to a certain tolerance.

ASM using rigorous mathematical analysis. There are different ways to realize an OSM. Here we define the OSM by introducing two unknowns along the interface, Γ, instead of u_Γ. We denote them by λ₁,λ₂ and introduce the following two conditions

γA_Γλ₁ + (1−γ)A_Γλ₂ + A^>_1Γu₁ + A^>_2Γu₂ = f_Γ,

(1−γ)A_Γλ₁ + γA_Γλ₂ + A^>_1Γu₁ + A^>_2Γu₂ = f_Γ, (1.5) where ¹₂ < γ ≤ 1. Note that if we subtract these two conditions we arrive at λ₁ = λ₂ =u_Γ. Therefore the following augmented system







A₁ A_1Γ

A^>_1Γ γAΓ A^>_2Γ (1−γ)AΓ

A₂ A_2Γ A^>_1Γ (1−γ)A_Γ A^>_2Γ γA_Γ











 v₁ λ₁ v₂ λ2





=





 f₁ f_Γ f₂ f_Γ





 (1.6)

has the same solution as (1.4) in the sense that v₁ =u₁, v₂ = u₂ and λ₁ =λ₂ =u_Γ. We later useγto improve the convergence of the OSM. The OSM is defined by applying a block Jacobi method to (1.6) instead of (1.4): for a given initial guess,w⁽⁰⁾, solve for w^(m) until convergence using

M˜w^(m) = ˜Nw^(m⁻¹⁾+g,

where g := (f₁,f_Γ,f₂,f_Γ)^>, ˜N = ˜M −A˜ and ˜A is the augmented matrix in (1.6).

Here ˜M is the block diagonals of ˜A, that is

M˜ :=







A₁ A_1Γ A^>_1Γ γA_Γ

A₂ A_2Γ A^>_2Γ γA_Γ





. (1.7)

Let us now perform some numerical experiments on these iterative methods. Note that both methods can be run in parallel because both M and ˜M matrices are block diagonal. We are interested in the performance of ASM and OSM when we refine the grid, i.e., h→0 or equivalently n→ ∞.

For a given grid, we execute ASM and OSM by choosing a random initial guess for u⁽⁰⁾ and w⁽⁰⁾. We iterate until we reach a desired accuracy in the error, that is

kw−w^(m)k₂ ≤ tol₁, ku−u^(m)k₂ ≤ tol₂.

Then we measure the number of iterations required to reach this tolerance and observe how it behaves as we refine the grid size, i.e.,h→0. We choosef = 1 and the number of unknowns ranges from 360 to 408,960 from the coarsest to the finest grid.

In Table 1.1 we observe that the OSM with γ = ¹₂(1 +√

h) requires the minimum number of iterations compared to the two other methods. Moreover the number of

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iterations for this method grows likeh⁻^1/2 compared toh⁻¹ for the other two methods.

This is a desired property for an iterative method, i.e., the number of iterations depends less on the size of the linear system compared to other methods. In order to understand the surprising fast convergence of the OSM with optimal γ we need to address some natural questions that arise from analyzing iterative methods:

1. Is OSM a well-posed algorithm? more precisely, can we show that ˜M is invertible?

2. Can we obtain a contraction factor for OSM with its dependence onh and choice of parameter γ?

3. How does OSM behave in the presence of many subdomains?

4. Does fast convergence happen for other discretizations, e.g., the discontinous Galerkin method?

5. Can we use OSM as a preconditioner for a Krylov method?

We will address the above questions in this thesis. For the first one we will show that ˜M is s.p.d. if γ >1/2 and this holds for different discretizations; therefore ˜M is invertible. In order to answer the second question we need to study the spectral radius of the operatorρ( ˜M⁻¹A). We will derive a contraction factor which depends on˜ handγ.

Then we show that the optimal performance is obtained ifγ = ¹₂(1 +√

h). We address the third question by defining a many subdomain algorithm and show its convergence rate. The analysis of the convergence rate relies on certain estimates that hold for different discretizations and therefore we have a positive answer to the forth question.

Finally we show that ˜M is a good preconditioner for the matrix ˜A with optimal γ but as we already see ˜A is a non-symmetric matrix and therefore we use a Krylov method for non-symmetric matrices like GMRES. Unfortunately the convergence of GMRES is not only governed by ρ( ˜M⁻¹A) and we cannot obtain a sharp contraction factor.˜ But numerical experiments show that GMRES converges with the contraction factor ρ≤1−O(h^1/4) using the optimal γ.

1.2 Chapters summary

This thesis consists of two main parts: Chapter 2 (numerical methods for an elliptic problem) and Chapter 3 (domain decomposition methods). Chapter 2 is devoted to recall two main numerical methods for elliptic boundary value problems. Chapter 3 is the main contribution of this thesis and contains a short review of some iterative linear solvers along with a proposed solver and its analysis which is applicable to linear systems obtained from discretizations of elliptic PDEs.

• Chapter 2: We first introduce the Poisson problem in a weak form and review existence and uniqueness of the solution to the weak formulation in Section 2.1.

Then in Section 2.2 we show how we can approximate the exact solution of the above mentioned weak form. In this section we introduce classical finite element method (FEM) and discontinuous Galerkin (DG) discretization as approximation methods to the exact solution. In Section 2.3 we show how one can analyze both FEM and DG method in a unified abstract framework. In particular we will recall some assumptions which helps us in obtaining error estimates. Section 2.4

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and Section 2.5 are devoted to study in details different DG methods and their approximation properties. We conclude this chapter by performing some numerical experiments to validate (numerically) theoretical approximation properties of different DG methods in Section 2.6.

• Chapter 3: In Section 3.1, we recall how a block Jacobi method applied to the linear system of a DG discretization can be interpreted as a domain decomposition method at the continuous level. We then recall in Section 3.2 how analysis of such domain decomposition methods at the continuous level helps us in designing solvers for DG discretizations. In this section we propose the same solver introduced in Chapter 1 for a DG discretization and give some heuristic arguments on its performance. Section 3.3 can be considered as the core of this thesis where we rigorously analyze our solver and show its superior convergence behavior compare to classical solvers for a two subdomain configuration. We then generalize our analysis to many subdomain case in Section 3.4. Later on in Section 3.5 we explore other applications of our solver. Finally in Section 3.6 we show that our theoretical findings match with numerical experiments.

• Appendix: In Appendix A we recall and prove some technical estimates needed to obtain main results of Section 3.3 and Section 3.4. In Appendix B we can find a summary of thesis in French.

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Chapter 2

Numerical methods for an elliptic problem

We considered a Poisson problem as PDE in the introduction under restrictive conditions on the right-hand side and shape of the domain. In this chapter we restate the Poisson problem in a weak sense as our model problem which allows us to work with a more general right-hand side and more general domains. The analysis of elliptic problems, e.g., Poisson problem, requires extensive preliminaries from Sobolev spaces, functional analysis and measure theory. Since this thesis is not aimed at the theory of elliptic PDEs but rather at the numerical analysis of such PDEs, in Section 2.1 briefly recall the common notions for the analysis of elliptic PDEs and refer the reader to textbooks devoted to this subject.

The weak solution of the Poisson problem lives in an infinite-dimensional space and modern numerical methods, e.g., finite element methods or discontinuous Galerkin methods, seek an approximation in a finite-dimensional (sub)space. In Section 2.2 recall the Ritz-Galerkin approximation methods and state an abstract error analysis.

We then study in Section 2.4 how a discontinuous Galerkin approximation mimics the weak solution of the continuous problem.

2.1 Elliptic PDEs and weak solutions

Recall the Poisson equation with homogenous Dirichlet boundary condition

−∆u = f, in Ω⊂R²,

u = 0, on∂Ω, (2.1)

where ∆ := ^∂_∂x²^u2 +^∂_∂y²^u2 and derivatives are taken in the classical (strong) sense. Then a natural choice for the space of functions in which we should look for the solution is the space of twice differentiable functions, i.e., C²(Ω). Now suppose that the right-hand side, f, has a discontinuity inside the domain. Then (2.1) implies that u does not have a continuous second derivative at that point, i.e., u6∈C²(Ω). This motivates the need for a “weaker” notion of solution and definition of the PDE or in other words we require less regularity for the solution, u.

Suppose that the solution of (2.1) has continuous second derivatives, i.e.,u∈C²(Ω).

Now let us multiply the PDE by a test function, v, that vanishes on the boundary of the domain and is sufficiently smooth for the moment. We then integrate over the

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domain, use integration by parts (Green’s identity) to obtain Z

Ω

∇u· ∇v− Z

∂Ω

∂u

∂nv

| {z }

=0

= Z

Ω

f v. (2.2)

Note that the left-hand side of (2.2) is well-defined if the test function vanishes on the boundary and has continuous first derivatives; in fact it is enough that the integral on the left-hand side is bounded. The same is true for the right-hand side of (2.2).

We can ensure that the expression on the left-hand side is well-defined by applying Cauchy-Schwarz

Z

Ω

∇u· ∇v ≤ Z

Ω

|∇u|²

1/2 Z

Ω

|∇v|² 1/2

<∞,

which is bounded requiring that∇uand∇vare square (Lebesgue) integrable. Moreover we can use weak derivatives instead of the classical one. Similarly for the right-hand

side we have Z

Ω

f v ≤ Z

Ω

|f|²

1/2 Z

Ω

|v|² 1/2

<∞,

requiring that f and v are square (Lebesgue) integrable. In other words if f ∈ L²(Ω) where

L²(Ω) :=

n v :

Z

Ω

|v|² <∞o

, (2.3)

and u, v ∈H¹₀(Ω) where

H¹₀(Ω) :=

n

v :v,∂v

∂x,∂v

∂y ∈L²(Ω), v|∂Ω = 0 o

, (2.4)

then we can ensure that (2.2) makes sense. Note that for (2.2) to make sense we do not need to assume that f is continuous but rather just square (Lebesgue) integrable.

Similarly we do not need a very smooth solution. More precisely we can restate the

“strong” problem in (2.1) in a “weaker” form: Given f ∈ L²(Ω), find u∈ H¹₀(Ω) such

that Z

Ω

∇u· ∇v = Z

Ω

f v, ∀v ∈H¹₀(Ω). (2.5)

If (2.1) admits a smooth solution, u ∈ C²(Ω), then it coincides with the solution of (2.5); we integrate by parts back to the original PDE by assuming that u is twice differentiable. However there might be situations where (2.1) does not admit any solution while (2.5) does.

There is a natural question whether or not the weak formulation (2.5) has a solution and furthermore if it is unique. It turns out that we are looking for the solution in the right space, i.e., u∈H¹₀(Ω). In order to show existence and uniqueness of the solution we can use either the Riesz representation Theorem or the Lax-Milgram Lemma. The latter can be applied to a wider set of problems, e.g., advection-diffusion, and we use it here. Before stating the Lax-Milgram Lemma, let us rewrite the weak formulation in an abstract form: Givenhf,·i ∈H⁰, find u∈H such that

a(u, v) =hf, vi, ∀v ∈H, (2.6)

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where H is a Hilbert space and H⁰ is its dual space, a(·,·) : H×H → R is a bilinear form (linear in each argument) and hf,·i ∈ H⁰ is a linear bounded functional. The Lax-Milgram Lemma for this problem reads

Lemma 2.1 (Lax-Milgram) Let H be a Hilbert space and a(·,·) : H×H → R such that

|a(u, v)| ≤c₁kuk kvk, ∀u, v ∈H, (boundedness), (2.7) a(v, v)≥c₂kvk², ∀v ∈H, (coercivity), (2.8) where c₁, c₂ > 0 and k · k is the Hilbert space norm. Moreover Let hf,·i ∈ H⁰. Then there is a unique solution to the problem (2.6) and kuk ≤ck hf,·i k⁰.

In order to show that our weak formulation (2.5) has a unique solution, we need to show that the assumptions of the Lax-Milgram Lemma are satisfied. We let H = H¹₀(Ω) with the norm

kvk²₁ :=kvk²_Ω+k∇vk²_Ω, (2.9) where kvk_Ω := R

Ω|v|²1/2

and

a(u, v) :=

Z

Ω

∇u· ∇v. (2.10)

For the dual space we define

hf, vi:=

Z

Ω

f v, ∀v ∈H¹₀(Ω), (2.11)

and k hf,·i k⁰ := kfk_Ω. We need to show boundedness and coercivity of the bilinear form. The former is easy using the Cauchy-Schwarz inequality. For the latter we use the Poincar´e-Friedrichs inequality.

Theorem 2.2 (Poincar´e-Friedrichs inequality) Let Ω be a bounded domain with Lipschitz boundary. Then there exists a positive constant, C_Ω, which depends only on the shape of the domain such that for all v ∈H¹₀(Ω) we have

kvk_Ω ≤C_Ωk∇vk_Ω. (2.12)

Therefore a(v, v) = k∇vk²_Ω is a norm and satisfies the coercivity condition (2.8). The norm associated to a(·,·) is often calledenergy norm.

A natural question is how smooth a weak solution is given f ∈L²(Ω)? It is shown that provided the boundary of Ω is smooth enough (of class C² or Ω is a convex polygon), then u∈H²(Ω) where H²(Ω) is defined as

H²(Ω) :=

n

v :v ∈H¹(Ω),∂²v

∂x², ∂²v

∂x∂y, ∂²v

∂y∂x,∂²v

∂x²,∈L²(Ω) o

.

In two-dimension we have by Sobolev embedding theorem that H²(Ω) is a subspace of continuous functions. This property is used in designing finite-dimensional subspaces in finite element method which we discuss in next section.

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2.2 Approximation of the weak solution: Ritz-Galerkin method

Recall that the solution of the weak formulation (2.5) lives in an infinite dimensional space,u∈H¹₀(Ω). In general we do not know how to solve (2.5) and instead we look for an approximation of the solution. The Ritz-Galerkin method is an approach to search for the approximation in a finite dimensional space. A nice historical development of this approach is given in [30].

In the Ritz-Galerkin approximation we replace both the solution and test spaces by a finite dimensional space, say V_h. We denote the dimension of V_h by n := dim(V_h).

We then solve the following problem instead of (2.5): Given f ∈ L²(Ω) find u_h ∈ V_h such that

a(u_h, v) = hf, vi, ∀v ∈V_h. (2.13) If we choose a set of basis functions for V_h, i.e., V_h = span{φ₁, φ₂, . . . , φ_n}, then the variational problem is equivalent to a linear system

Au=f, (2.14)

where A ∈Rⁿ^×ⁿ is called system matrix and is defined by [A]_ij :=a(φ_i, φ_j), and u= (u₁, u₂, . . . , u_n)^> is the vector representing u_h in the given basis, i.e. u_h =Pn

i=1u_iφ_i. Similarly we have f = (f₁, f₂, . . . , f_n)^> where f_i :=hf, φ_ii for i= 1, . . . , n. Therefore solving this linear system leads to an approximation of the weak solution.

Note that we have not placed any constraint on the choice of the finite dimensional space, V_h. One option would be to choose a space which is a subspace of H¹₀(Ω), V_h ⊂ H¹₀(Ω). Since under some mild conditions u ∈ H²(Ω)∩H¹₀(Ω) and therefore the weak solution iscontinuous in two-dimensions, it is desirable to construct an approximation space which is continuous. The classical finite element method (FEM) is such an example. FEM is constructed by first triangulating the domain Ω, and we denote triangulation by Th. Then the FEM space is defined as the set of all continuous functions which are piecewise polynomial and vanish at the boundary of the domain, i.e.,

V_FEM:=

v :v ∈C(Ω), v|K∈Th ∈P^k(K), v|∂Ω = 0 , (2.15) wherekis a positive integer. HereP^k(K) is the space of polynomials of degree at mostk in an elementK ∈ Th, e.g., for the reference element ˆK := triangle

(0,0),(1,0),(0,1) we have

P¹( ˆK) := span

1, x, y , P²( ˆK) := span

1, x, y, x², xy, y² .

The diameter of a triangle K ∈ Th is denoted by h_K and usually in the literature h is defined as the maximum diameter of a triangulation. Note that if we uniformly refine the triangulation thenh→0. Ashtends to zero (as we have more and more triangles) the dimension ofV_h grows. This is why his used as a measure for the dimension of V_h in the analysis of finite element methods. We refer the reader to [9] for the construction and analysis of FEMs.

We do not have to restrict ourselves by choosing subspaces of H¹₀(Ω) as approximation spaces. The discontinuous Galerkin (DG) method which we will discuss extensively in Section 2.4 uses a discontinuous function as approximation to the continuous solution! The approximation space is defined as the set of all functions which are piecewise polynomial but not necessarily continuous, i.e.,

V_DG:=

v :v ∈L²(Ω), v|K∈Th ∈P^k(K) . (2.16)

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Relaxing the continuity of the approximation comes at a price. We need to modify also the bilinear form of the approximation: instead of a(·,·), DG mimics continuity of the continuous solution using a penalization technique; the jump across elements is penalized in the bilinear form.

The main question about the Ritz-Galerkin methods is whether they converge to the true solution and if yes under which conditions and in which sense.

2.3 Abstract error analysis

In this section we recall the convergence theory for Ritz-Galerkin methods. We would like to show that as the dimension of the approximation space grows, n → ∞, then u_h converges to u in some norm. In order to analyze both FEM and DG methods in the same framework, we suppose that the approximation is obtained from an abstract problem: Find u_h ∈V_h such that

a_h(u_h, v) = hf, vi, ∀v ∈V_h, (2.17) where ah(·,·) is a bilinear form which is not necessarily the same as a(·,·). Moreover V_h does not have to be a subspace of H¹₀(Ω). We assume that V_h, an abstract norm k · k, and a_h(·,·) satisfy the following conditions:

1. Boundedness: There exists a positive constant,c₁, such that|a_h(w, v)| ≤c₁kwk kvk for all w, v ∈V(h) :=V_h+ H¹₀(Ω)∩H²(Ω)

.

2. Coercivity: There exists a positive constant, c₂, such that a_h(v, v) ≥ c₂kvk² for all v ∈V_h. Note that we did not require coercivity on H¹₀(Ω) but on V_h.

3. Galerkin orthogonality: The continuous solution, u ∈ H¹₀(Ω) ∩H²(Ω), satisfies (2.17), i.e., a_h(u, v) = hf, vi for all v ∈V_h.

4. Adjoint consistency: Let ψ ∈H¹₀(Ω)∩H²(Ω) be the weak solution of

−∆ψ = g, in Ω,

ψ = 0, on ∂Ω. (2.18)

We say ah(·,·) is adjoint consistent ifah(v, ψ) =hg, vi for all v ∈V(h).

Remark 2.1 The FEM satisfies all the assumptions introduced here by definition since V_FEM ⊂H¹₀(Ω).

Remark 2.2 The symmetry and coercivity of the bilinear form leads to a symmetric positive-definite (s.p.d.) matrix. More precisely after choosing a basis function for V_h we will have a symmetric system matrix, i.e., A=A^>, which satisfies

v^>Av >0, ∀v ∈Rⁿ,v 6= 0.

This also shows that the linear system (2.14) is solvable and has a unique solution.

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The consequence of Galerkin orthogonality is that a_h(u−u_h, v) = 0 for allv ∈V_h. Let w∈V_h then we have

kw−u_hk² ≤ 1

c₂a_h(w−u_h, w−u_h), (coercivity)

= 1

c₂a_h(w−u+u−u_h, w−u_h),

= 1 c₂

a_h(w−u, w−u_h) +a_h(u−u_h, w| {z }−u_h

=:v∈Vh

)

,

= 1 c2

a_h(w−u, w−u_h), (Galerkin orthogonality)

≤ c₁

c₂kw−uk kw−u_hk. (continuity) Therefore we have

kw−u_hk ≤ c1

c₂kw−uk, ∀w∈V_h. (2.19) For the error, u−u_h, we use the triangle inequality

ku_h−uk=ku_h−w+w−uk ≤ ku_h−wk+kw−uk ≤ 1 + c1

c₂

kw−uk. Since w∈V_h is arbitrary, we conclude that

ku_h−uk ≤ 1 + c₁ c₂

inf

w∈V_hkw−uk. (2.20) which shows that our approximation is the best approximation up to a constant. This is called C´ea’s Lemma. Note that the question of how good our approximation is to the weak solution of the PDE is transformed to how well we can approximate a function in H¹₀(Ω)∩H²(Ω) by a function in Vh.

The best operator for approximating a function from H¹₀(Ω)∩H²(Ω) into V_h is a projection. However we choose a less optimal operator: interpolation. This is because it is easier to work with interpolation compared to projection and the cost of using interpolation is only a constant which is independent of the dimension of V_h, i.e., n.

Under some mild conditions on the triangulation which we state in Section 2.4.1 we can obtain an estimate for the error. Let h be the mesh parameter of a regular and quasi-uniform triangulation, Th. Ifu∈ H¹₀(Ω)∩H^m(Ω) , 2≤ k+ 1≤m then for both V_FEM and V_DG we have

winf∈Vh

kw−uk ≤Ch^k|u|_H^m_(Ω). (2.21) Note that as h →0 (as we refine the mesh), the dimension of V_h goes to infinity, i.e., n→ ∞. Substituting (2.21) into (2.19) proves convergence of the approximate solution to the weak solution:

ku−u_hk ≤Ch^k |u|_H^m_(Ω). (2.22) We have shown that an approximation obtained from (2.17) which satisfies certain conditions converges in energy norm. However we are also interested in obtaining convergence in other norms like the L²-norm. Recall that from adjoint consistency we have by setting g =u−uh

a_h(v, ψ) = hu−u_h, vi, ∀v ∈V(h).

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Now choosing, v = u−u_h ∈ V(h) we have a_h(u−u_h, ψ) = ku−u_hk²_Ω. By Galerkin orthogonality of the original problem we have a_h(u −u_h,I_hψ) = 0, where I_h is the interpolation operator. This gives

ku−u_hk²_Ω =a_h(u−u_h, ψ−I_hψ)≤c₁ku−u_hk kψ−I_hψk ≤Ch |ψ|_H²_(Ω) ku−u_hk, where in the last step we used the interpolation estimate argument again. If the boundary of domain is smooth enough we can use regularity results and conclude that

|ψ|_H²_(Ω) ≤cku−u_hk_Ω. Therefore for L²-norm of the error we obtain

ku−u_hk_Ω ≤Chku−u_hk ≤Ch^k+1|u|_H^m_(Ω), (2.23) which is a faster convergence rate compared to the energy norm. The above argument is called the Aubin-Nitsche trick.

2.4 Discontinuous Galerkin methods

So far we have analyzed approximation methods to the weak solution of our PDE in an abstract form. Assumptions and the analysis of Section 2.3 are straightforward for FEM method due to the simplicity of the approximation space, VFEM ⊂ H¹₀(Ω), and its bilinear form, i.e., ah(u, v) := a(u, v) = R

Ω∇u· ∇v. However this is not the case for the discontinuous Galerkin (DG) methods where the approximation space contains discontinuous functions and the bilinear form is different. This flexibility in choosing the bilinear form results in many different DG methods, for instance we can have a non-symmetric method for a symmetric problem.

In this section we introduce some DG methods. To do so we first introduce some notations widely used in the DG community and then define DG methods in two different but equivalent forms: the primal and mixed formulations. We then show many DG methods satisfy assumptions of Section 2.3 and therefore enjoy optimal approximation properties. We finally study a class of DG methods which are called hybridizable DG methods.

2.4.1 Notation

We now define necessary operators and function spaces needed to analyze DG methods.

We follow the notation in [6]. Let Th be a partition of the domain Ω constructed by non-overlapping triangles, {Ki}. Every edge of a triangle is either a part of ∂Ω, or an entire edge of a neighboring triangle; there is no hanging node. This is called a conforming triangulation. For an example see Figure 2.1. We denote the set of interior edges shared by two elements inTh by E⁰, that is

E⁰ :={e:e=∂K₁∩∂K₂,∀K₁, K₂ ∈ Th}. Similarly we define the set of boundary edges by

E^∂ :={e:e=∂K∩∂Ω,∀K ∈ Th},

and all edges by E := E^∂ ∪ E⁰. We denote the diameter of K ∈ Th by h_K :=

max_x,y_∈K¯ |x−y| and define the mesh parameter byh:= max_K_∈T_hh_K.

Suppose we have a sequence of triangulations, T1,T2, . . ., such that Ti is obtained from uniformly refining triangles of Ti−1. In other words the mesh parameter of this

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K₂ K₄

K₂ K₃ K₁

K₁

K₃

P e

Figure 2.1: An example of a conforming triangulation (left) and a non-conforming one (right). Note that for non-conforming triangulation we havee=∂K1∩∂K3 which is not an edge ofK1. The node P is called a hanging node.

sequence of triangulations goes to zero,h→0. We assume that there exists a positive constant, β, independent of the triangulations of the sequence, such that

β ≤ ρK

h_K, ∀K ∈ Ti,

where ρ_K is a ball inscribed within the element K. A sequence of triangulations with such a property is called a regular mesh. This condition eliminates the possibility that long and thin triangles to form. We further require that there exist a positive constant, independent of the triangulation, such that

α ≤ h_K

h , ∀K ∈ Ti.

Such a triangulation is called to be quasi-uniform. This enforce that the triangulation is refined uniformly and not locally. Ifeis an edge of an element, we denote the length of that edge byh_e. The quasi-uniformity of the mesh implies h≈h_K ≈h_e.

Recall the approximation space of a DG method V_DG :=

v ∈L²(Ω) : v|_K ∈P^k(K),∀K ∈ Th ,

which is not a subspace of H¹₀(Ω). Instead it is a subspace of a broken Sobolev space H^l(Th) := Y

K∈Th

H^l(K),

where H^l(K) is the usual Sobolev space in K ∈ Th and l is a positive integer. Since H^l(Th) contains discontinuous functions, its trace space alongE⁰ can be double-valued.

We define the trace space of functions in H^l(Th) by T(E) := Y

K∈Th

L²(∂K).

Observe thatq ∈T(E) can be double-valued on E⁰ but it is single-valued on E^∂. We now define two trace operators: let q ∈ T(E) and q_i := q|∂Ki. Then on e =

∂K₁∩∂K₂ we define the average and jump operators

{{q}}:= ¹₂(q₁+q₂), [[q]] := q₁n₁+q₂n₂,

wheren_iis the unit outward normal fromK_ione∈ E⁰. Note that the jump and average definition are independent of the elements enumeration. Similarly for a vector-valued function r ∈[T(E)]² we define on interior edges

{{r}}:= ¹₂(r₁+r₂), [[r]] :=r₁·n₁+r₂·n₂.

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On the boundary, we set the average and jump operators to {{r}} := r and [[q]] = qn and we do not need to define {{q}} and [[r]] on the boundary of the domain.

Since H^l(Th) contains discontinuous functions we need to define some piecewise gradient operators: for allv ∈H^l(Th) we define∇v to be the piecewise gradient in each element. For all u, v ∈H^l(Th) we define

Z

Ω

∇u· ∇v := X

K∈Th

Z

K

∇u· ∇v.

Fora, b∈T(E) and single-valued on E⁰ we define the edge integrals by Z

E

a b:=X

e∈E

Z

e

a b.

2.4.2 Primal and mixed formulations

We can now define different DG methods. There are two different but equivalent formulations which cover most DG methods: primal and mixed formulations. In the primal formulation we have a variational form with u_h as the sole approximation unknown (primal variable) while in the mixed formulation we also approximate the gradient of the PDEs unknown, ∇u. The mixed formulation is the analog of mixed methods for the classical finite element method.

Let us start by rewriting (2.1) as a system of first order PDEs σ =∇u, −∇ ·σ =f in Ω, u= 0 on ∂Ω.

We now introduce a triangulation of the domain, Th ={K}, and multiply the above system by appropriate test functions and integrate by part in an element K to obtain

Z

K

σ·τ =− Z

K

u∇ ·τ + Z

∂K

uτ ·n_K, Z

K

σ· ∇v = Z

K

f v+ Z

∂K

vσ·n_K,

wherevandτ are test functions andn_Kis the normalized outward vector of∂K. A DG approximation to this system in mixed form is defined by: Find (u_h,σ_h)∈V_DG×[V_DG]^d, where d= 2 is the dimension of the problem, such that for allK ∈ Th

Z

K

σ_h·τ =− Z

K

u_h∇ ·τ + Z

∂K

ˆ

u_hτ ·n_K, ∀τ ∈[V_DG]^d, (2.24) Z

K

σ_h· ∇v = Z

K

f v+ Z

∂K

vσˆ_h·n_K, ∀v ∈V_DG. (2.25) Here

ˆ

u_h : H^m(Th)×[H^m(Th)]^d→T(E), ˆ

σ_h : H^m(Th)×[H^m(Th)]^d→[T(E)]^d,

are called numerical fluxes and they are approximations of u and σ on all edges, E. The definition of the numerical fluxes, ˆuh and ˆσh, completes the definition of the DG

Analysis of Schwarz methods for discontinuous Galerkin discretizations

Thesis

Reference

Analysis of Schwarz methods for discontinuous Galerkin discretizations

Analysis of Schwarz methods for discontinuous Galerkin discretizations

ﻪﻣﺪﻘﻣ

Preface 1

Contents

Chapter 1

Introduction

1.1 A numerical experiment

1.2 Chapters summary

Chapter 2

Numerical methods for an elliptic problem

2.1 Elliptic PDEs and weak solutions

2.2 Approximation of the weak solution: Ritz-Galerkin method

2.3 Abstract error analysis

2.4 Discontinuous Galerkin methods

Preface ¹