An explicit formulation of the multiplicative Schwarz preconditioner

Efficient preconditioners can be defined from domain decompositions with overlapping when solving linear systems arising from a PDE discretization. Partitioning of the domain entails a corresponding block partitioning of the matrix. It is known that Schwarz's methods (additive and multiplicative) correspond to given splittings of the matrix. An explicit formulation of the splitting was only known for the additive situation. An explicit splitting associated with the multiplicative Schwarz iteration is provided. The advantage of considering the explicit formulation, when the iteration is used as a preconditioner for a Krylov method, is shown.