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We present a weighted BFBT approximation (w-BFBT) to the inverse Schurcomplement of a Stokes system with highly heterogeneous viscosity. When used aspart of a Schur complement-based Stokes preconditioner, we observe robust fastconvergence for Stokes problems with smooth but highly varying (up to 10 ordersof magnitude) viscosities, optimal algorithmic scalability with respect to meshrefinement, and only a mild dependence on the polynomial order of high-orderfinite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$).For certain difficult problems, we demonstrate numerically that w-BFBTsignificantly improves Stokes solver convergence over the widely used inverseviscosity-weighted pressure mass matrix approximation of the Schur complement.In addition, we derive theoretical eigenvalue bounds to prove spectralequivalence of w-BFBT. Using detailed numerical experiments, we discussmodifications to w-BFBT at Dirichlet boundaries that decrease the number ofiterations. The overall algorithmic performance of the Stokes solver isgoverned by the efficacy of w-BFBT as a Schur complement approximation and, inaddition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG)method, which we use to approximate the inverses of the viscous block andvariable-coefficient pressure Poisson operators within w-BFBT. Building on thescalability 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 30,000cores of TACC's Lonestar 5 supercomputer.

Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity