Introduction to Multigrid Methods Chapter 7: Elliptic equations and Sparse linear systems Gustaf S ¨ oderlind 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 S ¨ oderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2009-2010 Introduction to Multigrid Methods – p. 1/61
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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
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