AILU: a preconditioner based on the analytic factorization of the ellipticoperator

We investigate a new type of preconditioner for large systems of linear equ ations stemming from the discretization of elliptic symmetric partial diffe rential equations. Instead of working at the matrix level, we construct an analytic factorization of the elliptic operator into two parabolic factors and we identify the two parabolic factors with the LU factors of an exact b lock LU decomposition at the matrix level. Since these factorizations are n onlocal, we introduce a second order local approximation of the parabolic f actors. We analyse the approximate factorization at the continuous level an d optimize its performance, which leads to the new AILU (Analytic ILU) prec onditioner with convergence rate 1 - O(h(1/3)), where h denotes the mesh si ze. Numerical experiments illustrate the effectiveness of the new approach. Copyright (C) 2000 John Wiley & Sons, Ltd.