Iterative Solvers and Preconditioners for Fully-Coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems

Abstract

Finite element discretization of fully-coupled, incompressible flow problems with the classic mixed velocity-pressure interpolation produces matrix systems that render the best and most robust iterative solvers and preconditioners ineffective. The indefinite nature of the discretized continuity equation is the root cause and is one reason for the advancement of pressure penalty formulations, least-squares pressure stabilization techniques, and pressure projection methods. These alternatives have served as admirable expedients and have enabled routine use of iterative matrix solution techniques; but all remain plagued by exceedingly slow convergence in the corresponding nonlinear problem, lack of robustness, or limited range of accuracy. The purpose of this paper is to revisit matrix systems produced by this old mixed velocity-pressure formulation with two approaches: (1) deploying well-established tools consisting of matrix system reordering, GM-RES, and ILU preconditioning on modern architectures with substantial distributed or shared memory, and (2) tuning the preconditioner by managing the condition number using knowledge of the physical causes leading to the large condition number. Results obtained thus far using these simple techniques are very encouraging when measured against the reliability (not efficiency) of a direct matrix solver. Here we demonstrate routine solution for an incompressible flow problem using the Galerkin finite element method, Newton-Raphson iteration, and the robust and accurate LBB element. We also critique via an historical survey the limitations of pressure-stabilization strategies and all other commonly used alternatives to the mixed formulation for acceleration of iterative solver convergence. The performance of the new iterative solver approaches on other classes of problems, including fluid-structural interaction, multi-mode viscoelasticity, and free surface flow is also demonstrated.

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

“…In addition, some researchers have found that stabilized element formulations can lead to improved performance of iterative solvers, see e.g. [24].…”

An arbitrary Lagrangian–Eulerian formulation for the numerical simulation of flow patterns generated by the hydromedusa Aequorea victoria

“…In here A 22 and b 2 are zero and A 12 -A > 21 . Although the system matrix of (23) is indefinite due to zero diagonal block, recent results indicate that indefiniteness of the problem does not represent a particular difficulty and a recent review of the iterative methods for solving large saddle point problems may be found in [6,39,44]. However, due to the zero diagonal block resulting from the divergence-free constraint, an ILU(k) type preconditioner cannot be used directly for the saddle point problem.…”

Stabilization of Low-order Mixed Finite Elements for the Stokes Equations

“…This popularity results from factors such as local mass conservation for the lowest order conforming pair (piecewise linear, bilinear or trilinear C 0 velocities and piecewise constant pressures), simple and uniform data structures for the lowest equal order pair (piecewise linear, bilinear or trilinear C 0 velocities and pressures), and algebraic problems with manageable sizes and small bandwidths in three dimensions for both pairs. The latter is of paramount importance in engineering applications where geometry resolution requires very fine meshes and higher order elements can quickly lead to intractable algebraic problems in three space dimensions; see [27] for an example setting.To counteract the lack of LBB stability, low-order pairs are usually supplemented by stabilization or postprocessing procedures that remove spurious pressure modes. Unlike penalty methods (see [16,22,24,25]) for which the goal is to uncouple the pressure and velocity, stabilized methods aim to relax the continuity equation so as to allow application of LBB incompatible spaces.…”

“…This popularity results from factors such as local mass conservation for the lowest order conforming pair (piecewise linear, bilinear or trilinear C 0 velocities and piecewise constant pressures), simple and uniform data structures for the lowest equal order pair (piecewise linear, bilinear or trilinear C 0 velocities and pressures), and algebraic problems with manageable sizes and small bandwidths in three dimensions for both pairs. The latter is of paramount importance in engineering applications where geometry resolution requires very fine meshes and higher order elements can quickly lead to intractable algebraic problems in three space dimensions; see [27] for an example setting.…”