We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to cal culate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with small fill-in. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in 0(n+e) time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in 0(ne) time. We conjecture that the problem of finding a minimum ordering is NP-complete.

We de ne and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin nite element method using piecewise linear polynomials.

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