Efficient multigrid solvers for mixed-degree local discontinuous Galerkin multiphase Stokes problems

Abstract

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.

Figures (13)

• Figure 1: Comparing solution accuracy for the two main LDG approaches: equal-degree (far-left and white circles $\circ$) and mixed-degree (middle-left and filled circles $\bullet$).
• Figure 1: Regions of near-optimal smoother parameters for 2D steady-state standard-form Stokes problems (left column), steady-state stress-form Stokes problems (middle column), and unsteady vanishing-viscosity Stokes problems (right column), for each ${{\wp}} \in \{1, 2, 3\}$. Here, "near-optimal" means the multigrid iteration count is at most 10% above optimal, according to the predictions of two-grid local Fourier analysis, as outlined in \ref{['sec:lfa']} and detailed further in \ref{['app:lfa']}. The approximate centroid and bounding box of each region is indicated at the top of each panel.
• Figure 1: Multigrid solver performance for the single-phase steady-state standard-form Stokes problem considered in \ref{['sec:A']}. For each configuration (2D or 3D, polynomial degree ${{\wp}}$, and boundary condition type), the corresponding sequence of symbols plot the numerically determined speed $\eta$ on grid sizes $n \times n\,(\times\,n)$ where $n = 4,8,16,32,\ldots$, from left-to-right. Periodic, Dirichlet, and stress boundary conditions are denoted by , $\raisebox{-0.1ex}{$\bullet$}$, and $\raisebox{-0.1ex}{$\circ$}$, respectively.
• Figure 2: Regions of near-optimal smoother parameters for 3D steady-state standard-form Stokes problems (left column), steady-state stress-form Stokes problems (middle column), and unsteady vanishing-viscosity Stokes problems (right column), for each ${{\wp}} \in \{1, 2, 3\}$. Here, "near-optimal" means the multigrid iteration count is at most 10% above optimal, according to the predictions of two-grid local Fourier analysis, as outlined in \ref{['sec:lfa']} and detailed further in \ref{['app:lfa']}. The approximate centroid and bounding box of each region is indicated at the top of each panel.
• Figure 2: Multigrid solver performance for the single-phase steady-state stress-form Stokes problem considered in \ref{['sec:B']}. For each configuration (2D or 3D, polynomial degree ${{\wp}}$, and boundary condition type), the corresponding sequence of symbols plot the numerically determined speed $\eta$ on grid sizes $n \times n\,(\times\,n)$ where $n = 4,8,16,32,\ldots$, from left-to-right. Periodic, Dirichlet, and stress boundary conditions are denoted by , $\raisebox{-0.1ex}{$\bullet$}$, and $\raisebox{-0.1ex}{$\circ$}$, respectively.