Multi-level incomplete factorizations for the iterative solution of non-linear finite element problems

Several engineering applications give rise quite naturally to linearized FE systems of equations possessing a multi-level structure. An example is provided by geomechanical models of layered and faulted geological formations. For such problems the use of a multi-level incomplete factorization (MIF) as a preconditioner for Krylov subspace methods can prove a robust and efficient solution accelerator, allowing for a fine tuning of the fill-in degree with a significant improvement in both the solver performance and the memory consumption. The present paper develops two novel MIF variants for the solution of multi-level symmetric positive definite systems. Two correction algorithms are proposed with the aim of preserving the positive definiteness of the preconditioner, thus avoiding possible breakdowns of the preconditioned conjugate gradient solver. The MIF variants are experimented with in the solution of both a single system and a long-term quasi-static simulation dealing with a multi-level geomechanical application. The numerical results show that MIF typically outperforms by up to a factor 3 a more traditional algebraic preconditioner such as an incomplete Cholesky factorization with partial fill-in. The advantage is emphasized in a longterm simulation where the fine fill-in tuning allowed for by MIF yields a significant improvement for the computer memory requirement as well.