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Block versions of the Jacobi preconditioner can be derived by a
partitioning of the variables. If the index set
is partitioned as with the sets mutually
The preconditioner is now a block-diagonal matrix.
Often, natural choices for the partitioning suggest themselves:
- In problems with multiple physical variables per node,
blocks can be formed by grouping the equations per node.
- In structured matrices, such as those from partial differential
equations on regular grids, a partitioning can be based on the
physical domain. Examples are a partitioning along lines in the
2D case, or planes in the 3D case. This will be discussed
further in §.
- On parallel computers it is natural to let the
partitioning coincide with the division of variables
over the processors.