AILU: a preconditioner based on the analytic factorization of the elliptic operator

AILU: A Preconditioner Based on the Analytic Factorization of the Elliptic Operator Martin J. Gander and Frederic Nataf Department of Mathematics, McGill University, Montreal, C anada and CMAP, CNRS UMR7641, Ecole Polytechnique, Palaiseau, France We investigate a new type of preconditioner for large system of linear equations stemming from the discretization of elliptic symmetric partial differential equations. Ins tead of working at the matrix level, we construct an analytic factorization of the elliptic operator into two parabolic f actors and we identify the two parabolic factors with the LU factors of an exact block LU decomposition at the matrix le ve . Since these factorizations are nonlocal, we introduce a second order local approximation of the parabol ic factors. We analyze the approximate factorization at the continuous level and optimize its performance which l eads to the new AILU (Analytic ILU) preconditioner with convergence rate 1 O(h1=3) whereh denotes the mesh size. Numerical experiments illustrate th e effectiveness of the new approach.