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NOTATION 8
Wp k (G)
H k (G)
H0 1(G)
H −1 (G)
N
Z
R
Sobolev space of functions with k-th derivatives in L p (G).
= W2 k (G).
the closure of C∞ 0 (G) in H1 (G).
the dual space of H0 1(G).
The natural numbers.
The integer numbers.
The real numbers.
Norms
‖q‖ 0 = ‖q‖ 0,2 for q ∈ L 2 (G).
|q| 1 = ‖∇q‖ 0 for q ∈ H 1 (G).
‖q‖ 1 = √ ‖q‖ 2 0 + ‖∇q‖2 0 for q ∈ H1 (G).
‖v‖ X = √ v T Xv for v ∈ R n and a symmetric positive definite matrix X ∈ R n×n .
‖v‖ l2 = √ v T v for v ∈ R n .
‖Y v‖
‖Y ‖ α = sup
α
0≠v∈ n
‖v‖ α
for Y ∈ R n×n
(consistent matrix norm to the vector norm ‖.‖ α ).
‖Y ‖ F =
√ ∑n
i,j=1 y2 ij for Y ∈ Rn×n (Frobenius norm).
Often used Indices, etc.
d
L
l
D
C
R
S
s
Space dimension.
Total number of multigrid levels.
Index indicating a certain multigrid level (l ∈ {1, . . . , L}).
Index indicating a diffusive term or Laplacian.
Index indicating a convective term.
Index indicating a reaction term.
Index indicating a stabilization term.
Index used if we want to emphasize that some operator is scalar.