A fast multigrid solver is presented for high-order accurate Stokes problems discretized by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an elementwise block Gauss–Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilization 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 and (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 multiphase embedded interface problems containing density and viscosity discontinuities several orders in magnitude; and test cases with curved geometries using semiunstructured meshes.

针对局部不连续伽勒金（LDG）方法离散化的高阶准确斯托克斯问题，提出了一种快速的多网格求解器。多重网格算法由一个简单的V周期组成，使用元素块高斯-赛德尔平滑器。这种方法的有效性取决于LDG压力损失稳定参数-只要选择合适的参数，数值实验表明（i）对于稳态Stokes问题，多网格求解器的收敛速度可以与经典几何多网格方法相匹配。对于泊松问题和（ii）对于非定常斯托克斯问题，随着有效雷诺数的增加，收敛速度进一步加快。广泛的二维和三维测试问题证明了求解器的性能以及高阶精度-这些问题包括周期性，Dirichlet和应力边界条件；包含密度和粘度不连续性几个数量级的可变粘度和多相嵌入式界面问题；使用半非结构化网格的弯曲几何形状的测试用例。