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, multi-coloured 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 to reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.

翻译：本文针对椭圆界面及多相斯托克斯问题，设计并研究了适用于高阶局部间断伽辽金方法的多种多重网格求解器。基于标准多重网格V循环框架，我们考察了多种单元块松弛算子，包括雅可比、多色高斯-赛德尔、处理器块高斯-赛德尔方法，并特别关注基于稀疏近似逆（SAI）方法的松弛算子。我们重点开发的SAI方法具有以下特点：（i）在斯托克斯问题中平衡速度与压力变量的平滑效果；（ii）稳健处理多相问题中的高对比度黏度系数。通过涵盖泊松方程、椭圆界面问题、稳态斯托克斯问题及非稳态斯托克斯问题的二维与三维测试案例，我们系统评估了多种多重网格松弛算子与求解器的组合方案。在所有测试案例中，至少存在一种方法的性能可与经典几何多重网格算法相媲美（例如通过4至8次迭代将残差降低10个数量级）。此外，我们还从算法简洁性、鲁棒性、计算成本及并行化潜力等方面对比了各方法的相对优势。