Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts.BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large‐scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single‐level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large‐scale saddle‐point problems, which often arise from systems of PDEs. We introduce HILUCSI, or Hierarchical Incomplete LU‐Crout with Scalability‐oriented and Inverse‐based dropping. As a multilevel preconditioner, HILUCSI statically and dynamically permutes individual rows and columns to the next level for deferred factorization. Unlike ILUPACK, HILUCSI applies symmetric preprocessing techniques at the top levels but always uses unsymmetric preprocessing and unsymmetric factorization at the coarser levels. The deferring combined with mixed preprocessing enabled a unified treatment for nearly or partially symmetric systems and simplified the implementation by avoiding mixed 1×1 and 2×2 pivots for symmetric indefinite systems. We show that this combination improves robustness for indefinite systems without compromising efficiency. Furthermore, to enable superior efficiency for large‐scale systems with millions or more unknowns, HILUCSI introduces a scalability‐oriented dropping in conjunction with a variant of inverse‐based dropping. We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier–Stokes equations. We also compare its performance with ILUPACK and the supernodal ILUTP in SuperLU.

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