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 disjoint, then

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.

Mon Nov 20 08:52:54 EST 1995