Symmetric Interior Penalty Discontinuous Galerkin Discretizations and Block Preconditioning for Heterogeneous Stokes Flow

Provable stable arbitrary order symmetric interior penalty (SIP) discontinuous Galerkin discretizations of heterogeneous, incompressible Stokes flowutilizing $Q^2_k$--$Q_{k-1}$ elements and hierarchical Legendre basis polynomials are developed and investigated. For solving the resulting linear system, a block preconditionediterative method is proposed. The nested viscous problem is solved by a $hp$-multilevel preconditioned Krylov subspacemethod. For the $p$-coarsening, a two-level method utilizing element-block Jacobi preconditionediterations as a smoother is employed. Piecewisebilinear ($Q^2_1$) and piecewiseconstant ($Q^2_0$) $p$- coarse spacesare considered. Finally, Galerkin $h$-coarsening is proposed and investigated for the two $p$- coarse spacesconsidered. Through a number of numerical experiments, we demonstrate that utilizing the $Q^2_1$ coarse spaceresults in the most robust $hp$- multigrid methodfor heterogeneous Stokes flow. Using this $Q^2_1$ coarse spacewe observe that the convergen...