Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier-Stokes Equations

Abstract

Monolithic preconditioners applied to the linear systems arising during the solution of the discretized incompressible Navier-Stokes equations are typically more robust than preconditioners based on incomplete block factorizations. Lower number of iterations and a reduced sensitivity to parameters like velocity and viscosity can significantly outweigh the additional cost for their setup. Different monolithic preconditioning techniques are introduced and compared to a selection of block preconditioners. In particular, two-level additive overlapping Schwarz methods (OSM) are used to set up monolithic preconditioners and to approximate the inverses arising in the block preconditioners. GDSW-type (Generalized Dryja-Smith-Widlund) coarse spaces are used for the second level. These highly scalable, parallel preconditioners have been implemented in the solver framework FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos. The new GDSW-type coarse space GDSW* is introduced; combining it with other techniques results in a robust algorithm. The block preconditioners PCD (Pressure Convection-Diffusion), SIMPLE (Semi-Implicit Method for Pressure Linked Equations), and LSC (Least-Squares Commutator) are considered to various degrees. The OSM for the monolithic as well as the block approach allows the optimized combination of different coarse spaces for the velocity and pressure components, enabling the use of tailored coarse spaces. The numerical and parallel performance of the different preconditioning methods for finite element discretizations of stationary as well as time-dependent incompressible fluid flow problems is investigated and compared. Their robustness is analyzed for a range of Reynolds and Courant-Friedrichs-Lewy (CFL) numbers with respect to a realistic problem setting.

Figures (19)

• Figure 4.1: Visualization of three types of (overlapping) interface decompositions in a $2\times 2\times 2$ domain decomposition.Left: GDSW interface components: 6 edges, 1 vertex, and 12 faces. Center: GDSW$^\star$ interface components: 1 vertex-based component (union of vertex and 6 edges), 12 faces. Right: RGDSW interface component: 1 vertex-based component (union of vertex, 6 edges, 12 faces).
• Figure 6.1: Test case 1: Backward-facing step (BFS) problem.Left: Backward-facing step geometry with structured mesh partition into 9 subcubes, each with side length 1. The subcubes correspond to the subdomains, and in the case displayed, we have $H/h=3$. For a higher number of processor cores, the subcubes are divided up further. For example, for 243 processor cores, each subcube holds $3\times 3 \times 3$ subdomains. Right: Solution to transient BFS problem at $t=\qty{10.0}{\s}$, using time step size $\Delta t = \qty{0.05}{\s}$. Reynolds number is 3 200 with kinematic viscosity $\nu=\qty[per-mode = symbol]{6.25e-4}{\cm^2\per\s}$. P2--P1 discretization with $H/h=9$, computed on 243 cores.
• Figure 6.2: Number of coarse functions for different number of processor cores (which equals the number of subdomains) for the backward-facing step (BFS) problem. The first name corresponds to the interface partition of unity for the velocity component and the second for the pressure component.
• Figure 6.3: Test case 2: Realistic artery problem. Geometry taken from Artery_Balzani_2012. Left: Magnitude of solution $\mathbf{u}_h$ at $t=\qty{0.8}{\s}$ with initial maximum inflow velocity $v_{\max}=\qty[per-mode=symbol]{40}{\cm\per\s}$. Right: Inflow profile over time of the realistic artery problem. Ramp phase until $t=\qty{0.1}{\s}$ and subsequent initial maximum inflow velocity $v_{\max}=\qty[per-mode=symbol]{40}{\cm\per\s}$. Maximum Reynolds number calculated based on maximum velocity and on approximate maximum diameter of artery; see \ref{['Def: Re artery max']}. The heartbeat and the corresponding flow rate were constructed in balzani:2015:nmf based on inflow pressure data from Hemolab.
• Figure 6.4: Comparison of different coarse space combinations for the velocity (first component) and pressure (second component) for a P1--P1 discretization and the monolithic preconditioner. Average GMRES iteration count per Newton step. Total number of Newton iterations required to reach convergence is 5. Weak scaling test. Stationary BFS problem with $H/h=17$. $\textrm{Re}=200$. Pressure projection is not used. Left: Excluding off-diagonal blocks in $\phi$ to build the coarse matrix. Right: Using full $\phi$ to build the coarse matrix.