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CHAPTER 3. MULTIGRID METHODS 42
Thus
Λ ≤ 2 ω
∑
j,k
j≠k
h jk (e 2 j + e 2 k) − 2 ∑ j,k
j≠k
h − jk (e2 j + e 2 k). (3.12)
With e = (0, . . . , 0, 1, 0, . . . , 0) T , the i-th unit vector, we get from (3.10)
∑
h ij ≤ ω ∑ h − ij ,
j
j
j≠i
j≠i
which together with (3.12) gives
and therefore completes the proof.
Λ ≤ 0
From now on we will write
A ≥ B
for two matrices A and B if A − B is positive semi-definite (or A > B if it is positive
definite), e.g. we can express (3.11) as 2 D ω H ≥ H.
We shortly sketch the construction of a reasonable PC
F for an essentially positive type
matrix H l = (h ij ) ij according to [Stü01a]. The construction is done in a way that for a
coarse level vector e C the interpolation PC F e C “fits smoothly” to e C , i.e. that if we set
( )
P
F
e = CI e C
then
h ii e i + ∑ j∈N i
h ij e j ≈ 0, for i ∈ F , (3.13)
where N i is the set of neighboring F- and C-nodes of F-node i, i.e.
N i := {j : j ≠ i, h ij ≠ 0}
the direct neighborhood. We will denote the subset of N i with negative matrix connections
with N − i , and P i ⊆ C ∩ N − i will be the set of interpolatory nodes, i.e. the set of C-nodes
which prolongate to F-node i. If we assume that for smooth error e
1 ∑
1 ∑
∑ h ij e j ≈ ∑
h ij e j
j∈P i
h ij
j∈P i
j∈N i
h ij
j∈N i
we could approximate (3.13) by
h ii e i + κ i
∑
j∈P i
h ij e j = 0,