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

TL;DR

This paper develops and analyzes stable high-order symmetric interior penalty discontinuous Galerkin discretizations for variable viscosity Stokes flow, introducing a block preconditioned iterative solver with multilevel techniques that demonstrate robustness against viscosity jumps.

Contribution

It introduces a novel stable SIP discretization for variable viscosity Stokes flow using hierarchical basis polynomials and proposes a robust block preconditioned solver with multilevel coarsening strategies.

Findings

Q^2_1 coarse space yields the most robust multigrid method.

Solver convergence is robust to viscosity jumps and mildly depends on polynomial order.

Theoretical analysis confirms the importance of penalty parameter choice based on local viscosity.

Abstract

Provable stable arbitrary order symmetric interior penalty discontinuous Galerkin (SIP) discretisations of variable viscosity, incompressible Stokes flow utilising $Q^2_k$--$Q_{k-1}$ 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 twolevel method utilising 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 utilising the $Q^2_1$ coarse space results in the most robust $hp$-multigrid method for variable viscosity Stokes flow. Using this $Q^2_1$ coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order $k$. It is demonstrated and supported by theoretical results that the convergence of the SIP discretisations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity.