Nonoverlapping domain decomposition:: A linear algebra viewpoint

In this work we consider the Helmholtz equation in a hyperparallelepiped Ω ⊂ R d , d = 1, 2, 3, …, under Dirichlet boundary conditions and for its solution we apply the averaging technique of the nonoverlapping Domain Decomposition, where Ω is decomposed in two, in general not equal, subdomains. Unlike what many researchers do that is first to determine regions of convergence and optimal values of the relaxation parameters involved at the PDE level, next discretize and then solve the linear system yielded using the values of the parameters determined, we determine regions of convergence and optimal values of the parameters involved after the discretization takes place, that is at the linear algebra level, and then use them for the solution of the linear system. In the general case the parameters obtained in this work are not the same with the ones which are known and which have been obtained at the PDE level.