Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 250

u ,k, 1
• • u ,k, 2

•

u ,k, 2

•

u
Fig. 53. The coarse-grid correction as an orthogonal projection

In addition, it follows from the orthogonality and (3.5) that

u ,k, 2

2

= u ,k, 2

2

+ u ,k, 2 − u ,k, 2

2

− u − u

+ ρ 2 µ

≤ u ,k, 2

2

u ,k, 1 − u ,k, 2

2

− u

− 1

− ρ 2 µ + ρ 2 µ

1 ) u ,k, 2

2

u ,k, 1

2 .

= ( 1 − u − u 3 . 8 )

− − 1

Now we make use of our knowledge of the two-grid rate. By (3.2),

− ρ 2 µ

1 )ρ 1 + ρ 2 µ

u ,k, 2

2

u ,k

2 .

− u ≤ [ ( 1

1 ]

− u

− −

Thus, (3.1) holds with a rate which can be estimated by (3.9).

3.3 A Recurrence Formula. The multigrid method with µ = 1 for the V-cycle

and µ = 2 for the W-cycle satisfies

ρ 2 µ ρ 2 ρ 1 + ρ 1 ) ≤

1 ( 1

− ( 3 . 9 )

−

at level ≥ 2 with respect to the energy norm.

1

3.4 Theorem. If the two-grid rate with respect to the energy norm satisfies ρ 1 ≤

2 ,

then

6
5 ρ 1 ≤

ρ ≤ 0 . 6 , for = 2 , 3 ,... ( 3 . 10 )

for the W-cycle.
Proof. For = 1 there is nothing to prove. By the assertion for − 1, it follows

from the recurrence formula (3.9) that

≤ ρ 1 + ( 6 + ( 6 ρ 2

5 ρ 1 ) 4 ( 1

− ρ 1 ) = ρ 1 {

5 ) 4 [ ρ 1 ( 1

− ρ 1 ) ] 1 }

+ ( 6

1
4

3
4 }≤

36

≤ ρ 1 {

5 ) 4 25 ρ 1 ≤

1 0 . 36 .

Taking the square root, we get the desired result.