Symmetric interior penalty discontinuous Galerkin discretisations and block preconditioning for heterogeneous Stokes flow - Charrier, D; May, D; Schnepp, S (Society for Industrial and Applied Mathematics, 2017)
Provable stable arbitrary order symmetric interior penalty discontinuous Galerkin (SIP) discretisations of heterogeneous, incompressible Stokes flow utilising Q2 k –Qk−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 (Q2 1 ) and piecewise constant (Q2 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 Q2 1 coarse space results in the most robust hp-multigrid method for heterogeneous Stokes flow. Using this Q2 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.