Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity

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 × P disc k−1 , order k ≥ 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. *

Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies

“…For the three‐dimensional problem, we use elements on nonconforming hexahedral meshes (Elman et al., 2014; Rudi et al., 2017). The dual stress variable is treated pointwise at quadrature nodes and not approximated by using finite elements.…”

“…Hence, linear iterative methods and preconditioners for large‐scale computational geophysics are important and rich research topics. The development and implementation of fast and scalable parallel linear solvers for three‐dimensional Stokes problems have been addressed by us (Isaac et al., 2015; Rudi, 2018, Rudi et al, 2015, 2017) and others (Fraters et al., 2019; May & Moresi, 2008; May et al., 2015). In this paper, however, we focus on nonlinear solvers and refer the reader to the above literature for numerically solving the linearized systems.…”

Pragmatic Solvers for 3D Stokes and Elasticity Problems with Heterogeneous Coefficients: Evaluating Modern Incomplete LDL<sup>T</sup> Preconditioners

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“…Specialized, optimal solvers often lack robustness as problem parameters or problem types are varied, though progress is being made in this regard (e.g. Rudi et al, 2017).…”

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

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