To avoid unnecessary complications, we first confine our discussion to this operator, to homogeneous Dirichlet conditions and to continuous, piecewise linear finite elements.

In the classical formulation of Schwarz's alternating method, two overlapping subregions, and Q; are used; the union of the two is Q. There are two sequential, fractional steps in which the approximate solution of the elliptic equation on Q is updated by solving the problem restricted to one of the subregions, one at a time. The most recent values of the solution are used as boundary values on the part of 392, the boundary of 92, which is not a part of 39.

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