1. Geometric MultiGrid Mehod
MULTIGRID METHOD
MultiGrid
Methods
(MGM)
Algebraic MultiGrid
Method
(AMG)
Geometric MultiGrid Mehod
(GMG)

2. MULTIGRID METHOD Remark: Given an approximate:
We want to improve this approximate:
Residual Equation:
It means that if we replace the right hand side vector b in the main linear system, the solution of this new system is the exact error
(residual)
(error)
But the cost of solving this system is the same as solving the original system.
(update)

3. MULTIGRID METHOD (GMG)
Two-Grid Cycle Algorithm
1) few GS iterations on the system
to obtain an approximate solution
Pre-smoothing step
2) Compute the residual
3) Restrict the residual
Restriction step
4) Solve residual equation on coarse mesh
ProloCorrection step
5) Interpolate
Prolongation step
6) Update
7) few GS iterations on the system
With initial approximate
Post-smoothing step

4. MULTIGRID METHOD (GMG)
W - Cycle
postsmoothing
restriction
presmoothing
prolongation
V - Cycle
Direct solver

5. MULTIGRID METHOD (GMG)
Jacobi code
function [sol]=jacobi(A,b,x0,steps)
n=length(b);
x=x0;
xnew = sparse(n,1);
for k=1:steps
for i=1:n
sum1=0;
for j=1:i-1
sum1=sum1+A(i,j)*x(j);
end
sum2=0;
for j=i+1:n
sum2=sum2+A(i,j)*x(j);
xnew(i)=(b(i)-sum1-sum2)/A(i,i);
x=xnew;
sol=xnew;

6. MULTIGRID METHOD (GMG)
one-dimensional problems
Example:
Coarse mesh
Fine mesh
We have the inclusion

7. MULTIGRID METHOD (GMG)

8. MULTIGRID METHOD (GMG)

9. MULTIGRID METHOD (GMG)
This means that the j-th column of P, corresponding to coarse grid node j, has the form

10. Two Grid TWO GRID METHOD nf=2*55-1; % number of fine nodes
h=1/(nf+1); % fine mesh size
xh=0:h:1; % fine mesh
oneh=ones(nf,1);
Ah=spdiags([-oneh 2*oneh -oneh],-1:1,nf,nf)/h; % fine stiffness matrix
bh=ones(nf,1)*h; % fine load vector
nc=(nf-1)/2; % coarse nodes
H=1/(nc+1); % coarse mesh size
xH=0:H:1; % coarse mesh
oneH=ones(nc,1);
AH=spdiags([-oneH 2*oneH -oneH],-1:1,nc,nc)/H; % coarse stiffness matrix
uh=zeros(nf,1); % initial guess
P=sparse(nf,nc); % prolongation matrix
for i=1:nc
P(2*i-1,i)=0.5; P(2*i,i)=1; P(2*i+1,i)=0.5;
end
R=P'; % Restriction matrix
for k=1:5
[uh]=jacobi(Ah,bh,uh,2);
rH=R*(bh-Ah*uh); % Restricted residual
eH=AH\rH; % correction step
uh =uh+P*eH; % update
plot(xh,[0 ; uh ; 0]), xlabel('x'), ylabel('u'); grid on
Two Grid

11. TWO GRID METHOD nf=2*55-1; % number of fine nodes
h=1/(nf+1); % fine mesh size
xh=0:h:1; % fine mesh
oneh=ones(nf,1);
Ah=spdiags([-oneh 2*oneh -oneh],-1:1,nf,nf)/h; % fine stiffness matrix
bh=ones(nf,1)*h; % fine load vector
nc=(nf-1)/2; % coarse nodes
H=1/(nc+1); % coarse mesh size
xH=0:H:1; % coarse mesh
oneH=ones(nc,1);
AH=spdiags([-oneH 2*oneH -oneH],-1:1,nc,nc)/H; % coarse stiffness matrix
uh=zeros(nf,1); % initial guess
P=sparse(nf,nc); % prolongation matrix
for i=1:nc
P(2*i-1,i)=0.5; P(2*i,i)=1; P(2*i+1,i)=0.5;
end
R=P'; % Restriction matrix
for k=1:5
[uh]=jacobi(Ah,bh,uh,2);
rH=R*(bh-Ah*uh); % Restricted residual
eH=AH\rH; % correction step
uh =uh+P*eH; % update
plot(xh,[0 ; uh ; 0]), xlabel('x'), ylabel('u'); grid on

12. MULTIGRID METHOD (GMG)
multigrid V-cycle is given by the following recursive algorithm: [Elman, Howard C., David Silvester, and Andy Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, 2014.] page102
u = V_cycle(A,b,u,level)
few GS iterations on the system
Pre-smooth
If coarsest level :
solve
else
Restriction
Recursive coarse grid correction
Prolong & update
endif
few GS iterations on the system
V - Cycle
Post-smooth

13. MULTIGRID METHOD (GMG)
u = V_cycle(A,b,u,level)
few GS iterations on the system
Pre-smooth
If coarsest level :
solve
else
Restriction
V – Cycle
3 level
Recursive coarse grid correction
Prolong & update
endif
few GS iterations on the system
Post-smooth

14. MULTIGRID METHOD (GMG)
clear; clc
kk=6; n1=4*kk+3;
% --- linear system for level 1
h1=1/(n1+1); x1=0:h1:1;
oneh1=ones(n1,1);
A1=spdiags([-oneh1 2*oneh1 -oneh1],-1:1,n1,n1)/h1;
b1=ones(n1,1)*h1;
% --- linear system for level 2
n2=(n1-1)/2; h2=1/(n2+1); x2=0:h2:1;
oneh2=ones(n2,1);
A2=spdiags([-oneh2 2*oneh2 -oneh2],-1:1,n2,n2)/h2;
b2=ones(n2,1)*h2;
% --- linear system for level 3
n3=(n2-1)/2; h3=1/(n3+1); x3=0:h3:1;
oneh3=ones(n3,1);
A3=spdiags([-oneh3 2*oneh3 -oneh3],-1:1,n3,n3)/h3;
b3=ones(n3,1)*h3;
A(1).size = n1; A(1).h = h1; A(1).mat = A1; A(1).rhs = b1;
A(2).size = n2; A(2).h = h2; A(2).mat = A2; A(2).rhs = b2;
A(3).size = n3; A(3).h = h3; A(3).mat = A3; A(3).rhs = b3;
b = b1; u = sparse(length(b),1);
level =3;
[u] = V_Cycle(A(1:level),b,u,level);
V – Cycle
3 level

15. MULTIGRID METHOD (GMG)
function [u] = V_Cycle(A,b,u,level)
if level < 1; return ; end
disp(['I am entering level ',num2str(level)])
if size(A,2) == 1
AA = A(1).mat;
u = AA\b;
disp(' ---coarset level done ---');
else
[u]=jacobi(A(1).mat,b,u,3); % (3 pre-smoothing)
nf = A(1).size; nc = A(2).size; P=sparse(nf,nc);
for i=1:nc
P(2*i-1,i)=0.5; P(2*i,i)=1; P(2*i+1,i)=0.5;
end
R=P';
rbar = R*(b-AA*u); % (restrict residual)
ebar = sparse(length(rbar),1);
Abar = A(2:level);
% recursive coarse grid correction ----
[ebar] = V_Cycle(Abar,rbar,ebar,size(Abar,2));
u = u + P*ebar; % prolong and update
[u]=jacobi(A(1).mat,b,u,3);
disp(['I am exiting level ',num2str(level)])
u = V_cycle(A,b,u,level)
Pre-smooth
few GS iterations on the system
If coarsest level :
solve
else
Restriction
Recursive coarse grid correction
Prolong & update
endif
few GS iterations on the system
Post-smooth

16. Geometric MultiGrid Mehod
MULTIGRID METHOD
MultiGrid
Methods
(MGM)
Algebraic MultiGrid
Method
(AMG)
Geometric MultiGrid Mehod
(GMG)

17. MULTIGRID METHOD (AMG)
Algebraic MultiGrid
Method
(AMG)
Can we apply multigrid techniques when there is no grid?
For AMG, these components are required.
sequence of grids,
P and R
smoothing
coarse-grid versions of the fine-grid operator
a solver for the coarsest grid.

18. MULTIGRID METHOD (AMG)
Algebraic MultiGrid
Method
(AMG)
Adjacency Graph
Selecting the Coarse Grid
we want to partition the grid points into C-pints and F-points.
Denote by Si the set of points that strongly influence i;
denote by ST the set of points that strongly depend on the point i.

19. MULTIGRID METHOD