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

We investigate a new type of preconditioner for large systems of linear equations stemming from the discretization of elliptic symmetric partial differential 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 block LU decomposition at the matrix level. Since these factorizations are nonlocal, we introduce a second order local approximation of the parabolic factors. We analyse the approximate factorization at the continuous level and optimize its performance, which leads to the new AILU (Analytic ILU) preconditioner with convergence rate 1 - (13), where denotes the mesh size. Numerical experiments illustrate the effectiveness of the new approach.