An efficient band-oriented scheme for solving n by n grid problems

Let <i>N</i> = <i>n</i>² for some positive integer <i>n</i>, and consider a square <i>n</i> by <i>n</i> grid consisting of (<i>n</i>--1)² small squares and having a node at each of the <i>n</i>² grid points. In this paper we consider the problem of directly solving the class of <i>N</i> by <i>N</i> symmetric positive definite linear systems of equations <i>Ax=b</i>, (1) where each <i>x</i>_<i>i</i> is associated with a grid point and A has the property that <i>A</i>_<i>ij</i>≠0 only if <i>x</i>_<i>i</i> and <i>x</i>_<i>j</i> are associated with nodes belonging to the same small square. We must specify how the unknowns are to be numbered if the above remark is to precisely determine the structure of <i>A</i>.