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Block Jacobi Methods

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.