A comparative study of efficient multigrid solvers for high-order local discontinuous Galerkin methods: Poisson, elliptic interface, and multiphase Stokes problems

We design and investigate efficient multigrid solvers for multiphase Stokes problems discretised via mixed-degree local discontinuous Galerkin methods. Using the template of a standard multigrid V-cycle, we develop a smoother analogous to element-wise block Gauss-Seidel, except the diagonal block inverses are replaced with an approximation that balances the smoothing of the velocity and pressure variables, factoring in the unequal scaling of the various Stokes system operators, and optimised via two-grid local Fourier analysis. We evaluate the performance of the multigrid solver across an extensive range of two- and three-dimensional test problems, including steady-state and unsteady, standard-form and stress-form, single-phase and high-contrast multiphase Stokes problems, with multiple kinds of boundary conditions and various choices of polynomial degree. In the lowest-degree case, i.e., that of piecewise constant pressure fields, we observe reliable multigrid convergence rates, though not especially fast. However, in every other case, we see rapid convergence rates matching those of classical Poisson-style geometric multigrid methods; e.g., 5 iterations reduce the Stokes system residual by 5 to 10 orders of magnitude.