Alternating-Direction Incomplete Factorizations

To solve the system of linear equations $Aw = r$ that arises from the discretization of a two-dimensional selfadjoint elliptic differential equation, iterative methods employing easily computed incomplete factorizations, $LU = A + B$, are frequently used. Dupont, Kendall and Rachford [SIAM J. Numer. Anal., 5 (1968), pp. 559–573] showed that, for the DKR factorization, the number of iterations (arithmetic operations) required to reduce the A -norm of the error by a factor of $\varepsilon $ is $O(h^{ - {1 / 2}} \log ({1 / \varepsilon }))(O(h^{ - {5 / 2}} \log ({1 / \varepsilon })))$, where h is the stepsize used in the discretization. We present some error estimates which suggest that, if a pair of alternating-direction DKR factorizations are used, then the number of iterations (arithmetic operations) may be decreased to $O(h^{ - {1 / 3}} \log ({1 / \varepsilon }))(O(h^{ - {7 / 3}} \log ({1 / \varepsilon })))$. Numerical results supporting this estimate are included.