A class of linear solvers based on multilevel and supernodal factorization

The solution of large and sparse linear systems is a critical component of modern science and engineering simulations. Iterative methods, namely the class of modern Krylov subspace methods, are often adopted to solve large-scale linear systems. To improve the robustness and the convergence rate of the iterative methods, preconditioning techniques are often considered crucial components of the linear systems solution. In this thesis, a class of algebraic multilevel solvers is presented for preconditioning general linear systems equations arising from computational science and engineering applications. They can produce sparse patterns and save memory costs by employing recursive combinatorial algorithms. Robustness is enhanced by combining the factorization with recently developed overlapping and compression strategies, and by using efficient local solvers. We have shown the good performance of the proposed strategies with numerical experiments on realistic matrix problems, also in comparison against some of the most popular algebraic preconditioners in use today.