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 flow utilizing $Q^2_k$--$Q_{k-1}$ elements and hierarchical Legendre basis polynomials are developed and investigated. For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a $hp$-multilevel preconditioned Krylov subspace method. For the $p$-coarsening, a two-level method utilizing element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear ($Q^2_1$) and piecewise constant ($Q^2_0$) $p$-coarse spaces are considered. Finally, Galerkin $h$-coarsening is proposed and investigated for the two $p$-coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilizing the $Q^2_1$ coarse space results in the most robust $hp$-multigrid method for heterogeneous Stokes flow. Using this $Q^2_1$ coarse space we observe that the convergen...

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