Fast multigrid solution of high-order accurate multi-phase Stokes problems

TL;DR

This paper introduces a fast multigrid solver for high-order accurate multi-phase Stokes problems discretized by LDG methods, demonstrating efficiency and high-order accuracy across various complex test cases.

Contribution

The paper presents a simple V-cycle multigrid algorithm with an element-wise block Gauss-Seidel smoother for high-order LDG discretized Stokes problems, showing competitive convergence rates.

Findings

Convergence rate matches classical geometric multigrid for steady problems

Accelerated convergence with increasing Reynolds number in unsteady problems

Effective for complex geometries and multi-phase problems with discontinuities

Abstract

A fast multigrid solver is presented for high-order accurate Stokes problems discretised by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an element-wise block Gauss-Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilisation parameter -- provided the parameter is suitably chosen, numerical experiment shows that: (i) for steady-state Stokes problems, the convergence rate of the multigrid solver can match that of classical geometric multigrid methods for Poisson problems; (ii) for unsteady Stokes problems, the convergence rate further accelerates as the effective Reynolds number is increased. An extensive range of two- and three-dimensional test problems demonstrates the solver performance as well as high-order accuracy -- these include cases with periodic, Dirichlet, and stress boundary conditions; variable-viscosity and multi-phase embedded interface problems containing density and viscosity discontinuities several orders in magnitude; and test cases with curved geometries using semi-unstructured meshes.