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With ¬nite difference approximations of PDEs, it is standard to block the variables

and the matrix by partitioning along whole lines of the mesh. For example, for the two-

dimensional mesh illustrated in Figure 2.5, this partitioning is

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This corresponds to the mesh 2.5 of Chapter 2, whose associated matrix pattern is shown

in Figure 2.6. A relaxation can also be de¬ned along the vertical instead of the horizontal

lines. Techniques of this type are often known as line relaxation techniques.

In addition, a block can also correspond to the unknowns associated with a few con-

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secutive lines in the plane. One such blocking is illustrated in Figure 4.2 for a grid. ¦ §

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The corresponding matrix with its block structure is shown in Figure 4.3. An important

difference between this partitioning and the one corresponding to the single-line partition-

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ing is that now the matrices are block-tridiagonal instead of tridiagonal. As a result,

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solving linear systems with may be much more expensive. On the other hand, the num-

ber of iterations required to achieve convergence often decreases rapidly as the block-size

increases.

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Matrix associated with the mesh of Figure 4.2.

Finally, block techniques can be de¬ned in more general terms. First, by using blocks

that allow us to update arbitrary groups of components, and second, by allowing the blocks

to overlap. Since this is a form of the domain-decomposition method which will be seen

later, we de¬ne the approach carefully. So far, our partition has been based on an actual

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set-partition of the variable set into subsets , with the

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condition that two distinct subsets are disjoint. In set theory, this is called a partition of . ¦

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