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Provable stable arbitrary order symmetric interior penalty discontinuousGalerkin (SIP) discretisations of variable viscosity, incompressible Stokesflow utilising $Q^2_k$--$Q_{k-1}$ elements and hierarchical Legendre basispolynomials are developed and investigated.For solving the resulting linearsystem, a block preconditioned iterative method is proposed. The nested viscousproblem is solved by a $hp$-multilevel preconditioned Krylov subspace method.For the $p$-coarsening, a twolevel method utilising element-block Jacobipreconditioned 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, wedemonstrate 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 solverappears to be robust with respect to the jump in the viscosity and only mildlydepending on the polynomial order $k$. It is demonstrated and supported bytheoretical results that the convergence of the SIP discretisations and theiterative methods rely on a sharp choice of the penalty parameter based onlocal values of the viscosity.

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