Domain decomposition for elliptic partial differential equations with Neumann boundary conditions (Q1107285)

A domain decomposition oriented version of an incomplete block- factorization preconditioner is studied numerically. The matrix arises from cell-centered finite difference approximation of the Neumann problem for an elliptic self-adjoint equation. This domain decomposition preconditioner is constructed as usual by partitioning the region into a square mesh into boxes. The preconditioned conjugate gradient method is used to handle the problem. For each iteration one needs to solve a system with the preconditioning matrix, which involves first one iteration of the so-called MINV method, proposed by \textit{P. Concus}, \textit{G. H. Golub} and \textit{G. Meurant} [SIAM J. Sci. Stat. Comput. 6, 220-252 (1985; Zbl 0556.65022)]. The 2nd step of the preconditioning is a modification of the residual on the interface boundary by use of the FFT [as used by \textit{J. H. Bramble}, \textit{J. E. Pasciak} and \textit{A. H. Sebatz} [Math. Comput. 47, 103-134 (1986; Zbl 0615.65112)]. The preconditioning concludes by one more iteration of the MINV- preconditioner in the subregions. Numerical experiments on the CRAY-XMP are presented.