We design and investigate a variety of multigrid solvers for high-order local
discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes
problems. Using the template of a standard multigrid V-cycle, we consider a variety
of element-wise block smoothers, including Jacobi, multicoloured Gauss–Seidel,
processor-block Gauss–Seidel, and with special interest, smoothers based on sparse
approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i)
balance the smoothing of velocity and pressure variables in Stokes problems; and (ii)
robustly handles high-contrast viscosity coefficients in multiphase problems. Across a
broad range of two- and three-dimensional test cases, including Poisson, elliptic
interface, steady-state Stokes, and unsteady Stokes problems, we examine a
multitude of multigrid smoother and solver combinations. In every case, there is at
least one approach that matches the performance of classical geometric multigrid
algorithms, e.g., 4 to 8 iterations reduce the residual by 10 orders of magnitude. We
also discuss their relative merits with regard to simplicity, robustness, computational
cost, and parallelisation.
