Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

Abstract

In this report we explore the performance of the SIMPLER , augmented Lagrangian, 'grad-div' preconditioners and their variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem Reynolds number varies from a low to the limiting number for a laminar flow.Mathematics subject classification: 65F10, 65F08.

Efficient and robust Schur complement approximations in the augmented Lagrangian preconditioner for the incompressible laminar flows

“…The main goal of this paper is to utilize the above mentioned Schur complement approximations to construct a new approximation of the Schur complement in the AL preconditioner, with more details presented in the next section. Theoretical analysis and numerical evaluation of the above Schur complement approximations fall out of the scope of this work and we refer to [1,3,34] for more results. Here we summarize the key differences.…”

“…which do not contain the coupling between two components of the velocity so that standard algebraic multigrid methods can be applied [34]. This advantage motivates us to choose P M AL in this paper despite the observation that the performance of P M AL is dependent of the values of γ , which is seen in the numerical experiments of this paper and other related references [14].…”

“…Alternatively, previous work [12] suggests to solve the systems with A γ by the Krylov subspace methods, which are accelerated by the approximate inverse preconditioner based on the Shermon-Morrison formula. In the related work [34], the comparison between the modified and ideal AL preconditioners is realized by applying the direct solution method for the involved sub-systems. Although fewer Krylov iterations are needed by the ideal AL preconditioner, removing the difficulty to solve the sub-systems with A γ makes the modified AL preconditioner attractive in practice.…”

Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

“…In fact, recent developments in this direction were tested in low to moderate Re-setups and the reported numbers indicate an unfavourable scaling of the iteration numbers with Re; see, e.g., [ESW05, Ch. 8] or [HV16]. Sometimes, in particular when it comes to parallelization, Gauss-Seidel -type methods are used, which iteratively and locally update the variables.…”

A low-rank update for relaxed Schur complementpreconditioners in fluid flow problems

“…Such block preconditioners typically require approximations to the inverse of the upper left block A of the system matrix in (1) as well as the (negative) pressure Schur complement S := BA −1 B T . Well-known preconditioning techniques for the Schur complement are SIMPLE-type preconditioners [24,13], the least-squares commutator (LSC) [9,10], the pressure-convection-diffusion commutator [17], grad-div preconditioners [7,19,11], augmented Lagrangian preconditioners [8,13,14] as well as hierarchical matrix preconditioners [18]. An alternative approach that is not based on the block LU factorization is given by the so-called nullspace method [6].…”