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Introduction to Multigrid Methods
Chapter 7: Elliptic equations and Sparse linear systems
Gustaf Soderlind
Numerical Analysis, Lund University
Textbooks: A Multigrid Tutorial, by William L Briggs. SIAM 1988
A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles. Cambridge 1996
Matrix-based multigrid: Theory and Applications, by Yair Shapira. Springer 2008
Multi-Grid Methods and Applications, by Wolfgang Hackbusch, 1985
c© Gustaf Soderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2009-2010
Introduction to Multigrid Methods – p. 1/61
1. Overview
Consider simplest linear 2p-BVP (1D Poisson)
y′′ = f(x)
y(0) = α; y(1) = β
Introduce equidistant grid with ∆x = 1/(N + 1)
Discretization
yi+1 − 2yi + yi−1
∆x2= f(xi)
y0 = α; yN+1 = β
Introduction to Multigrid Methods – p. 2/61
Linear system of equations
Tridiagonal N ×N matrix formulation
1
∆x2
−2 1
1 −2 1. . .
1 −2
y1
y2
...
yN
=
f(x1)− α/∆x2
f(x2)...
f(xN)− β/∆x2
In matrix–vector form
T∆xy = f
Introduction to Multigrid Methods – p. 3/61
Elliptic model problem: 2D Poisson equation
∂2u
∂x2+
∂2u
∂y2= f(x, y)
on Ω = [0, 1]× [0, 1] with Dirichlet conditions u = 0 on ∂Ω
Uniform grid xi, yjN,Ni,j=1, mesh width ∆x = ∆y = 1/(N + 1)
Discretization Finite differences with ui,j ≈ u(xi, yj)
ui−1,j − 2ui,j + ui+1,j
∆x2+
ui,j−1 − 2ui,j + ui,j+1
∆y2= f(xi, yj)
Introduction to Multigrid Methods – p. 4/61
Equidistant mesh ∆x = ∆y
ui−1,j + ui,j−1 − 4ui,j + ui,j+1 + ui+1,j
∆x2= f(xi, yj)
Participating approximations and mesh points
xi−1 xi xi+1
yj
yj+1
yj−1
Introduction to Multigrid Methods – p. 5/61
Computational “stencil” for ∆x = ∆y
ui−1,j + ui,j−1 − 4ui,j + ui,j+1 + ui+1,j
∆x2= f(xi, yj)
“Five-point operator”1
1 −4 1
1
Introduction to Multigrid Methods – p. 6/61
The FDM linear system of equations
Lexicographic ordering of unknowns⇒ partitioned system
1
∆x2
T I 0 . . .
I T I
I T I
. . . I
. . . 0 I T
u·,1
u·,2
u·,3
...
u·,N
=
f(x·, y1)
f(x·, y2)
f(x·, y3)...
f(x·, yN )
with Toeplitz matrix T = tridiag(1 −4 1)
The system is N 2 ×N 2, hence large and very sparse
Introduction to Multigrid Methods – p. 7/61
3D Poisson equation. The “curse” of dimension
Partitioned system
1
∆x2
T I 0 . . . I
I T I. . .
I T I
I. . . I
. . . 0 I T
u1,1,1
...
...
...
uN,N,N
=
f1,1,1
...
...
...
fN,N,N
with Toeplitz matrix T = tridiag(1 −6 1)
The system is N 3 ×N 3, hence extremely large and sparse
Introduction to Multigrid Methods – p. 8/61
Galerkin method (Finite Element Method)
1. Basis functions ϕi
2. Approximate u =∑
cjϕj
3. Determine cj from∑
cj
∫
∇ϕi · ∇ϕj =∫
fϕi
The cj are determined by the linear system
Kc = F
The stiffness matrix K has similar structure to FDM matrix
Stiffness matrix elements kij =∫
∇ϕi · ∇ϕj = a(ϕi, ϕj)
Right-hand side Fi =∫
ϕif = 〈ϕi, f〉 =∑
fj〈ϕi, ϕj〉
Introduction to Multigrid Methods – p. 9/61
The FEM mesh. Domain triangulation
Piecewise linear basis ϕj with triangulation mesh
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Same number of nodes too
Introduction to Multigrid Methods – p. 10/61
Iterative methods
All discretizations lead to very large linear systems
Ty = f
typically having millions of equations
Matrix factorization methods are out of the question! Useiterative methods instead!
Explicit iterative methods ym+1 = Bym + c
Implicit iterative methods Dym+1 = Bym + c
Introduction to Multigrid Methods – p. 11/61
What are multigrid methods?
Multigrid methods are iterative methods that use the factthat the origin of the linear system is some discretization,and that the grid properties affect the convergence rate
There is a relation to Fourier analysis as it turns out thatmesh width (inverse spatial frequency) is a key factorgoverning convergence
The methods are called multigrid, because the iteration willalternate between several different grids in order to speedup convergence
Introduction to Multigrid Methods – p. 12/61
2. Iterative methods for linear systems
Given a linear system Au = f construct sequence um → u
Then
Aum = f + rm
Au = f
Definitions
1. The error is defined by em = um − u
2. The residual is defined by rm = Aum − f
3. Relation via the error–residual equation Aem = rm
Introduction to Multigrid Methods – p. 13/61
The basic iterative methods
There are four different basic iterative methods
1. The Jacobi method
2. The Gauss–Seidel method
3. The Successive Overrelaxation (SOR) method
4. The Symmetric SOR method
Advanced methods include the Conjugate Gradient (CG)method; the Generalized Minimum Residiual (GMRES)method; and various forms of Multigrid (MG) methods
Introduction to Multigrid Methods – p. 14/61
The Jacobi method
Splitting Write Au = f as (D − L− U)u = f with
D diagonal; L lower triangular; U upper triangular
Du = (L + U)u + f
u = D−1(L + U)u + D−1f
Jacobi method Use fixed point iteration
um+1 = D−1(L + U)um + D−1f
Introduction to Multigrid Methods – p. 15/61
The Jacobi method. Implementation
Given um, calculate residual rm, and update according to
rm ← Aum − f
um+1 ← um −D−1rm
Note The scheme implies that each single, scalarequation is solved independently of the other equations
It can be directly used on massively parallel computers
Introduction to Multigrid Methods – p. 16/61
Relation to fixed point iteration
From actual implementation
um+1 = um −D−1((D − L− U)um − f) = D−1(L + U)um + D−1f
For analysis purposes this is written
um+1 = PJum + D−1f
where the Jacobi iteration matrix is
PJ = D−1(L + U)
Introduction to Multigrid Methods – p. 17/61
1D Poisson + Jacobi method
If A = T∆x then
PJ = D−1(L + U) = tridiag(1/2 0 1/2)
andum+1 = PJum + D−1f
For the exact solution
u = PJu + D−1f
Error recursion em+1 = PJem
Introduction to Multigrid Methods – p. 18/61
1D Poisson + Jacobi. Convergence
Error recursion em+1 = PJem
Convergence em → 0
1. Necessary condition ρ[PJ ] < 1
2. Sufficient condition ‖PJ‖ < 1
Definition Spectral radius ρ[A] = maxk |λk[A]|
For 1D Poisson, we need to calculate the eigenvalues ofthe Jacobi iteration matrix PJ = tridiag(1/2 0 1/2)
Introduction to Multigrid Methods – p. 19/61
Eigenvalues of Toeplitz matrices
PJ = tridiag(1/2 0 1/2) = S/2
Su =
0 1 0 . . .
1 0 1
1 0 1. . . 1
. . . 0 1 0
u1
u2
...
uN
= λu
Find the eigenvalues of S!
Introduction to Multigrid Methods – p. 20/61
Eigenvalues of symmetric Toeplitz matrix S
Consider the nth equation of Su = λu
un+1 + un−1 = λun
Linear difference equation with boundary values
u0 = 0; uN+1 = 0
Characteristic equation
z2 − λz + 1 = 0
Introduction to Multigrid Methods – p. 21/61
Eigenvalues of S. . .
Roots of z2 − λz + 1 = 0 are z and 1/z (product 1)
General solution un = αzn + βz−n
Boundary condition u0 = 0 = α + β ⇒
Solution un = α(zn − z−n)
Boundary conditionuN+1 = 0 = α(zN+1 − z−(N+1)) ⇒ z2(N+1) = 1
Roots zk = exp( kπi
N + 1
)
k = 1 : N
Introduction to Multigrid Methods – p. 22/61
Eigenvalues. . .
Sum of the roots of z2 − λz + 1 = 0 are
λk = zk + 1/zk ⇒
λk[S] = exp( kπi
N + 1
)
+ exp(
−kπi
N + 1
)
= 2 coskπ
N + 1
Hence the eigenvalues of PJ = tridiag(1/2 0 1/2) are
λk[PJ ] = coskπ
N + 1∈ (−1, 1)
Introduction to Multigrid Methods – p. 23/61
Eigenvalue locationsEigenvalues get very close to ±1 for large N (N = 31)
0 0.5 1 1.5 2 2.5 3−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1Eigenvalues cos(k*pi*dx) of Jacobi matrix
k*pi*dx, for k=1:N, with N=31
Eig
enva
lues
Introduction to Multigrid Methods – p. 24/61
Peacock plot. Eigenvalues and unit circle, N = 31
Eigenvalues λk = cos kπ∆x are projections on real axis
In terms of Chebyshev zeros, T ′
N+1(λk) = 0 and UN(λk) = 0
Introduction to Multigrid Methods – p. 25/61
Slow, slower, slowest
With λk[PJ ] = cos kπ/(N + 1) we have
ρ[PJ ] = λ1[PJ ] = −λN [PJ ]≈ 1−π2∆x2
2+ O(∆x4)
So Jacobi’s method will converge as ρ[PJ ] < 1, butconvergence will be painfully slow!
Example N = 99 ⇒ ρ[PJ ] ≈ 0.9995
Introduction to Multigrid Methods – p. 26/61
Convergence history
Plot of L2 error as a function of iteration number (N = 99)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010
−3
10−2
10−1
100
Convergence history
Iteration number
L2 E
rror n
orm
Introduction to Multigrid Methods – p. 27/61
Unit error reduction requires O(N2) iterations
Plot of L2 error after N 2 iterations (N = 15, 31, 63, 127)
0 2000 4000 6000 8000 10000 12000 14000 16000 1800010
−3
10−2
10−1
100
Convergence history: N=15, 31, 63, 127
Iteration number
L2 E
rror n
orm
Introduction to Multigrid Methods – p. 28/61
3. Power iteration
The fixed point iteration ym+1 = Aym is also called poweriteration
It implies ym = Amy0 Note powers of A!
Assume distinct eigenvalues Axk = λkxk with |λ1| > |λk|,
for k = 2, . . . , N . Let y0 = Σαkxk. Then
ym
λm1
= α1x1 +
N∑
k=2
αkxk
(
λk
λ1
)m
→ α1x1
So the vector ym gets aligned with eigenvector x1
Introduction to Multigrid Methods – p. 29/61
The power method
Once we have a good approximation to x1 we obtain thecorresponding eigenvalue from the Rayleigh quotient
λ1 ≈〈ym, Aym〉
〈ym, ym〉
The power method Take y0 and iterate until convergence:
ym := ym/‖ym‖2
ym+1 := Aym
σm := 〈ym, ym+1〉 → λ1
Introduction to Multigrid Methods – p. 30/61
Why study the power method?
The error recursion em+1 = PJem is a power iteration
Let PJv = λv, with |λ| = ρ[PJ ]. Then any convergent fixedpoint type iteration will have the following properties:
1. The error will initially decay relatively fast
2. Convergence then slows to be governed by ρ[PJ ]
3. The error em will become aligned with the eigenvector v
Introduction to Multigrid Methods – p. 31/61
What is the convergence to x1 like?
As
ym
λm1
= α1x1 +
N∑
k=2
αkxk
(
λk
λ1
)m
→ α1x1
we see that the error decay is exponential and governed bythe ratio of the largest to the next largest eigenvalue
The decay may be very slow if eigenvalue gap is small
For PJ the gap is very small, with ratio
λ2
λ1=
cos 2π∆x
cos π∆x≈
2− 4π2∆x2
2− π2∆x2≈ 1−
3π2∆x2
2
Introduction to Multigrid Methods – p. 32/61
What is the convergence to max |λ| like?
Non-normal matrices linear convergence
Normal matrices quadratic convergence
Example
A =
12 2 2.05
2 9 1
1.95 1 7
B =
12 2 2
2 9 1
2 1 7
Approximate eigenvalues λ ∈ 13.74 8.00 6.26, but thenormal matrix has orthogonal eigenvectors
Introduction to Multigrid Methods – p. 33/61
Computation. Non-normal vs. normal matrix
Convergence history: Error in λmax vs. iteration number
0 10 20 3010
−15
10−10
10−5
100
0 10 20 3010
−15
10−10
10−5
100
Introduction to Multigrid Methods – p. 34/61
Good news, bad news
Normal differential operator⇒ normal matrix
ut = uxx
ut = ∇ · (p∇u)
ut + ux = 0
ρt +∇ · (ρv) = 0
Non-normal differential operators
ut = ux +1
Peuxx
Introduction to Multigrid Methods – p. 35/61
4. Eigenvectors of PJ
From eigenvalue problem of PJ = tridiag(1/2 0 1/2)
Recall un = znk − z−n
k and zk = exp( kπiN+1
), then
ukn = exp
( knπi
N + 1
)
− exp(
−knπi
N + 1
)
∼ sin( knπ
N + 1
)
Same eigenvectors as for T∆x
Introduction to Multigrid Methods – p. 36/61
Eigenvectors of PJ at N = 31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Lowest mode, sin(n*pi*dx), on fine grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Highest mode, sin(n*N*pi*dx), on fine grid
Introduction to Multigrid Methods – p. 37/61
Eigenvectors of PJ at N = 31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Lowest mode, sin(n*pi*dx), on fine grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Highest mode, sin(n*N*pi*dx), on fine grid
Introduction to Multigrid Methods – p. 38/61
From fine grid to coarse – now you see it
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Lowest mode, sin(n*pi*dx), on fine grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Highest mode, sin(n*N*pi*dx), on fine grid
Introduction to Multigrid Methods – p. 39/61
From fine grid to coarse – now you don’t
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Lowest mode, sin(n*pi*dx), on coarse grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Highest mode, sin(n*N*pi*dx), on coarse grid
Introduction to Multigrid Methods – p. 40/61
And yet. . . Nyquist Sampling Theorem
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Lowest mode, sin(n*pi*dx), on coarse grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1Highest mode, sin(n*N*pi*dx), on coarse grid
Introduction to Multigrid Methods – p. 41/61
High frequency modes and grid density
What is a high frequency is a grid property
Nyquist Sampling Theorem On a grid with N interiorpoints the highest frequency that can be represented isun = sin Nπn/(N + 1)
Note
sinNπn
N + 1= sin
(
πn−πn
N + 1
)
= (−1)n+1 sin( πn
N + 1
)
Highest frequency is (−1)n modulation of lowest frequency!
Introduction to Multigrid Methods – p. 42/61
Downsampling and aliasing
Aliasing If we take only every second sample, the highestfrequency un = sin(Nπn/(N + 1)) “maps to” the function
vl = sin(2l + 1)π
N + 1∼ sin πx
or to the function
wl = − sin2lπ
N + 1∼ − sin πx
The former highest frequency maps to the lowest!
In general, if the Nyquist frequency of the coarse grid is ω
then a frequency ω + δω < 2ω “folds back” down to ω − δω
Introduction to Multigrid Methods – p. 43/61
Jacobi iteration demonstration
Error vector sequence (50 iterations) at N = 19
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07Error reduction in 50 iterations
Err
or
x
Introduction to Multigrid Methods – p. 44/61
5. The Lapacian and the 5-point FD operator
∂2u
∂x2+
∂2u
∂y2= λu
Computational domain Ω = [0, 1]× [0, 1] (unit square)Dirichlet conditions u(x, y) = 0 on boundary
Variable separation u(x, y) := v(x)w(y) implies
vxxw + vwyy = λvw
orvxx
v+
wyy
w= λ
Introduction to Multigrid Methods – p. 45/61
Eigenvalues and eigenfunctions of the Laplacian
Let v′′ = κkv and w′′ = µmw, then
λk,m = κk + µm
uk,m(x, y) = vk(x) · wm(y)
Modulation Theorem The Laplacian on the unit squarehas eigenvalues and eigenvectors, given by
λk,m = (k2 + m2)π2 k,m ∈ Z+
uk,m(x, y) = sin kπx sin mπy
Analogous modulation result in 3DIntroduction to Multigrid Methods – p. 46/61
The 5-point finite difference Laplacian
Equidistant mesh width ∆x = ∆y = 1/(N + 1)
Discretization Finite differences with ui,j ≈ u(xi, yj)
ui−1,j − 2ui,j + ui+1,j
∆x2+
ui,j−1 − 2ui,j + ui,j+1
∆x2= λ∆x
k,m · ui,j
Discrete variable separation uk,mi,j := vk
i · wmj implies
vki−1 − 2vk
i + vki+1
vki ∆x2
+wm
j−1 − 2wmj + wm
j+1
wmj ∆x2
= κ∆xk + µ∆x
m
Introduction to Multigrid Methods – p. 47/61
Discrete eigenvalues and eigenfunctions
Discrete Modulation Theorem The five-point finitedifference Laplacian on the unit square has N 2 discreteeigenvalues and eigenvectors, given by
λk,m = −4(N + 1)2 ·(
sin2 kπ
2(N + 1)+ sin2 mπ
2(N + 1)
)
uk,mi,j = sin
kπi
N + 1· sin
mπj
N + 1; i, j, k,m = 1 : N
Again analogous results hold in 3DThus 1D theory gives most of the necessary insight
Introduction to Multigrid Methods – p. 48/61
6. Advanced iteration methods
As Jacobi iteration is so slow, what can be done to speedup convergence?
Damped Jacobi (under-relaxation) Use a damping factorω and iterate according to
rm ← Aum − f
um+1 ← um − ωD−1rm
The iteration stops only when rm = 0, so it still solves theproblem Au = f
Introduction to Multigrid Methods – p. 49/61
Damped Jacobi method
Damped Jacobi is equivalent to
um+1 =(
(1− ω)I + ωD−1(L + U))
um + ωD−1f
so the iteration has damped Jacobi iteration matrix
Pω = (1− ω)I + ωPJ
Theorem The eigenvalues of Pω are
λk[Pω] = 1− ω + ω coskπ
N + 1
Introduction to Multigrid Methods – p. 50/61
Eigenvalue locations of damped Jacobi matrix
Eigenvalues for N = 31 and ω = 1, 0.5, 0.3, 0.2
0 0.5 1 1.5 2 2.5 3−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
k*pi*dx, for k=1:N, with N=31
Eig
enva
lues
Eigenvalues of damped Jacobi matrix
Introduction to Multigrid Methods – p. 51/61
Eigenvalues of damped Jacobi matrix
By choosing ω we can (at least) eliminate large negativeeigenvalues; what happens to the largest eigenvalue?
Largest eigenvalue becomes
λ1[Pω] = 1− ω + ω cosπ
N + 1= 1− 2ω sin2 π∆x
2
So
λ1[Pω] ≈ 1−ωπ2∆x2
2
(Ouch, this is even closer to 1 for ω < 1. . . )
Introduction to Multigrid Methods – p. 52/61
The Gauss–Seidel method
Splitting Write Au = f as (D − L− U)u = f with
D diagonal; L lower triangular; U upper triangular
(D − L)u = Uu + f
u = (D − L)−1Uu + (D − L)−1f
Gauss–Seidel method Use fixed point iteration
um+1 = (D − L)−1Uum + (D − L)−1f
Introduction to Multigrid Methods – p. 53/61
The Gauss–Seidel method. Implementation
Given um, calculate residual rm, and update according to
rm ← Aum − f
um+1 ← um − (D − L)−1rm
Note The scheme implies that each equation is solved insequence of scalar equations. It cannot be parallelized in astraightforward way
The Gauss–Seidel iteration matrix is
PGS = (D − L)−1U
Introduction to Multigrid Methods – p. 54/61
1D Poisson + Gauss–Seidel method
Major features compared to Jacobi iterations are
Regularizing iteration – no high-frequency error
Faster convergence, although still slow
Eigenvalues of PGS satisfy generalized eigenvalue problemUu = λ(D − L)u, according to Briggs:
λk[PGS ] = cos2 kπ
N + 1≈ 1− k2π2∆x2
but this result is incorrect!
Introduction to Multigrid Methods – p. 55/61
Gauss–Seidel eigenvalues and eigenvectors
[N/2] eigenvalues of PGS are
λk[PGS ] = cos2 kπ
N + 1≈ 1− k2π2∆x2
and the remaining eigenvalues are zero!
Note Eigenvectors of PGS and T∆x do not coincide!
Therefore we need other ways of studying convergence
Introduction to Multigrid Methods – p. 56/61
Digital filters and Bode diagrams
Study error recursion
em+1 = PGSem
one frequency at a time! Study “output” y = PGSu whenu = sin kπ∆x, an eigenvector of T∆x, and compute the L2
(root-mean-square) “attenuation”
‖y‖∆x
‖u‖∆x
=‖PGSu‖∆x
‖u‖∆x
=‖PGSu‖2‖u‖2
Frequency response Plot vs frequency ω = kπ∆x ∈ (0, π)
or as a function of wave number k = 1 : N (Bode diagram)Introduction to Multigrid Methods – p. 57/61
Bode diagram. Gauss–Seidel and Jacobi
Linear attenuation vs frequency: Gauss–Seidel and Jacobi
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (wave number)
Atte
nuat
ion
(line
ar)Frequency response: Guass−Seidel (b) and Jacobi (r)
Introduction to Multigrid Methods – p. 58/61
GS is a low-pass filter and J a band-stop filter
Logarithmic attenuation (dB): Gauss–Seidel and Jacobi
0 0.5 1 1.5 2 2.5 3−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
Frequency (wave number)
Atte
nuat
ion
(dB
)Frequency response: Guass−Seidel (b) and Jacobi (r)
Introduction to Multigrid Methods – p. 59/61
Gauss–Seidel as LP filter
Gauss–Seidel has good damping of all frequencies exceptlow frequencies which are only weakly suppressed
High frequency damping is better than 0.5, meaning thathigh frequency modes decay faster than 0.5m
Thus Gauss–Seidel is smoothing (LP filter) – the errorbecomes a smoother function as the iterations pass
As Jacobi does not damp high frequencies it is not asmoother; it blocks a narrow band of mid-frequencies
Introduction to Multigrid Methods – p. 60/61
Gauss–Seidel iteration demonstration
Error vector sequence (50 iterations) at N = 19
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04
−0.02
0
0.02
0.04
0.06
0.08
Introduction to Multigrid Methods – p. 61/61