Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity

TL;DR

This paper introduces a weighted BFBT preconditioner for Stokes problems with highly heterogeneous viscosity, demonstrating robust convergence, scalability, and improved solver performance through theoretical analysis and extensive numerical experiments.

Contribution

The paper develops a novel weighted BFBT approximation for the Schur complement in Stokes systems with highly variable viscosity, enhancing solver efficiency and scalability.

Findings

Achieves robust convergence for viscosities up to 10 orders of magnitude.

Demonstrates optimal scalability with respect to mesh refinement.

Significantly improves solver convergence over existing methods.

Abstract

We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer.