ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

Abstract

SUMMARYThe (mixed finite element) discretization of the linearized Navier-Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as 'grad-div' (GD) stabilization, as well as on the discrete side where it is known as an 'augmented Lagrangian' (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of H-LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of H-LU factorization in the preconditioners.

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

“…For the ideal and modified AL preconditioners, the number of linear iterations are close to each other in this report and [7,14] when the same settings are used, such as the same stopping tolerance, the same mesh and the same inner solver for the subsystems. For the ideal and modified 'grad-div' preconditioner, we compare our results with [9,15]. The same settings also result in a comparable result, in terms of the number of linear iterations.…”

“…There are many references testing the three preconditioners and their variants on the LDC problem, c.f. [7,14,9,15,16]. It is not easy to compare the results given in this report and other references, since different settings are used in different papers.…”

“…Adding a stabilization term −γ∇(∇·u) to the momentum equation in (1), one can obtain the so-called 'grad-div' stabilization formulation (cf. e.g., [9,15])…”

“…So far, the quite often used choices of W and γ has been the pressure mass matrix M (or diag(M )) and γ = 1 in [9,15]. In this report W = diag(M ) and γ = 1 are chosen.…”

“…It is clear that F GD is sparser than A AL since F GD arises in the discretization of an operator (also noted in [9]). On the other hand, G is analogous to the matrix B T B.…”