On methods of incomplete LU decompositions for solving Poisson's equation in annular regions (Q1181552)

Let an algebraic system \(Ax=b\) arise from a finite difference discretization of a self-adjoint elliptic partial equation and \(K\) be an approximation to the matrix \(A\): \(K=LU\), where factors \(L\) and \(U\) are constructed to have a similar sparsity pattern as the corresponding parts of \(A\). Since \(K\) is much easier to invert than \(A\) the iterative methods of the form \(x^{n+1}=x^ n+\alpha_ nK^{-1}(b-Ax^ n)\) is considered. Particular interest of the author is connected with the discretization of Poisson's equation in annular regions with a matrix \(A\) which has a periodic structure. In the paper two ways of incomplete \(LU\) decomposition process are discussed: 1) the method of \textit{J. A. Meierink} and \textit{H. A. van der Vorst} [J. Comput. Phys. 44, 134-155 (1981; Zbl 0472.65028)] and 2) the deletion and complementation method. The last one is the main topic and the development of the paper. It is shown, for example, that it is cheaper in terms of computational work than the first one and is also more robust.