Database Reference
In-Depth Information
Of course, this does no longer hold for multiple smoothing steps. For two
smoothing steps, we would obtain
x
:¼ x ^
ð
y
ÞþAx ^
ð
y
Þ Ax ^
ð
y
Þ¼x 2
y þ A
^
y
^
:
The resulting multigrid method is summarized in Algorithm 6.2. Hereupon, the
upper index 1 has been omitted for the sake of readability.
Algorithm 6.2: Multigrid V-cycle
Input: matrix A , right-hand side b , prolongator L , restrictor R , initial iterate
x
n
:¼ x 0
R
n of ( 6.4 )
x
e
R
Output: approximate solution
1: procedure VCYCLE( y )
2:
x : ¼ x y
pre-smoothing
y 1
3:
: ¼ R ( Ax y )
computing the residual
x 1
: ¼ ( A 1 ) 1 y 1
4:
direct solver on the coarse grid
x : ¼ x + Lx 1
5:
coarse grid correction
6:
x : ¼ x y
post-smoothing
7: return x
8: end procedure
9: return VCYCLE( b )
initial call
The following rather technical convergence result has been established in
[Pap10].
Theorem 6.1 (Theorem 3.7.1 in [Pap11]) Let
q ðÞ
j
α
,
i
,
j
G β
P
nn
R
, e
p ij
ð 6
:
15 Þ
,
0
else
for 0 α 1, and
X
q ðÞ
j
q ðÞ
j
0,
j
G β ,
¼ 1
, β∈
m
:
j∈G β
Moreover, we define
A
:¼ I γ
P ,
Q
:¼ I A ,
Þ 1 RA ,
K
:¼ I L RAL
ð
1
P P
ε :¼
,
k 1 , and
ε :¼ α
k
I RPL
q ðÞ
j
c
min
β , j
:
Search WWH ::




Custom Search