Line and Plane Separators

We consider sparse matrices arising from nite-element or nite-diierence methods. The graphs of such matrices are embedded in two or three dimensional Euclidean space, and the coordinates of their vertices are readily available. Such coordinate information was used earlier to develop a parallel Cartesian nested dissection heuristic that computes a ll reducing ordering for a matrix with an embedding in two dimensions. We extend Cartesian nested dissection to graphs embedded in three dimensions and to compute an ordering for a non-symmetric matrix A without explicitly forming the graph of A T A. We show that for an r-local graph with N vertices embedded in d dimensions (d=2, 3), a single step of Cartesian nested dissection computes a separator of size O(N 1?1=d). The separator also divides the graph into two subgraphs, each with at least N=5 vertices when d = 2 and at least N=7 vertices when d = 3. Computational results indicate that the algorithm performs rather well for a wide variety of graphs.