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CHAPTER 3. MULTIGRID METHODS 44
Choice of C nodes. What still has to be fixed is a concrete C/F-splitting. A very easy
to implement algorithm for this purpose is the red-black-coloring method [Kic98].
Algorithm 3.10. Red-Black Coloring
repeat until all the nodes are colored
begin
step 1: choose an uncolored node (e.g. with minimal node number);
step 2: this node is colored black;
step 3: all uncolored neighbors are colored red;
end
The black nodes are then used as C nodes.
A first variation of this algorithm is to use a different notion of ‘neighboring’ in step 3,
to color only the strongly negatively coupled (snc) nodes, where a node j is said to be snc
to a node k if
−h jk ≥ ε str max |h − ji |, (3.18)
i
with fixed parameter ε str ∈ (0, 1] (typically ε str = 0.25). We denote the set of strongly
negative couplings of a node j by
and the set of transposed strongly negative couplings by
Now step 3 can be replaced by
S j = {k ∈ N j : j is snc to k} (3.19)
S T j = {k : j ∈ S k }. (3.20)
“all uncolored nodes which are snc to the black node are colored red”. (3.21)
Another variant concerns step 1. The order in which the C nodes are chosen may be
crucial if we want to obtain a uniform distribution of C and F nodes. One suggestion in
this direction in [RS86] is to introduce a “measure of importance” λ j for each node j in
the set of ‘undecided’ nodes U, and to choose a node with maximal λ j as next C node.
One possibility for this measure is
λ j = |S T j ∩ U| + 2|ST j
∩ F |, (3.22)
which can be evaluated for all nodes in a preprocessing step and updated locally after each
iteration step.