In this study, a two-dimensional Geometric Multigrid (GMG) model for Poisson equation with interface on Structured Adaptive Mesh Refinement (SAMR) grid is developed. The model is designed to be combined with Navier-Stokes solvers with staggered arrangements of variables, although solvers with collocated arrangement are also applicable. Based on a general GMG method, special attention for flux conservation is taken on the coarse-fine interface. As large density ratios commonly exist in interface flows such as free surface flows, Galerkin Coarse grid Approximation (GCA) method is adopted to generate coefficients on coarse grids to enhance the robustness and efficiency of the model. Benchmark case is carried out for the validation of the model, and it is demonstrated to obtain 2nd-order accuracy and acceptable efficiency for even the density ratio reaches 10⁴. Point iteration and line relaxation iteration of Gauss-Seidel methods are compared to obtain a better performance. Furthermore, the model is implemented in a developed Navier-Stokes solver to validate its performance in simulating interface problems. The numerical results are compared with theoretical solution or experimental data from reliable sources.

在这项研究中，建立了带有结构自适应网格细化（SAMR）网格界面的Poisson方程的二维几何多重网格（GMG）模型。该模型设计为与变量交错排列的Navier-Stokes求解器组合，尽管也可以使用并置排列的求解器。基于通用的GMG方法，在粗细界面上要特别注意通量守恒。由于自由流动等界面流中普遍存在较大的密度比，因此采用Galerkin粗网格近似（GCA）方法在粗糙网格上生成系数，以提高模型的鲁棒性和效率。进行基准测试以验证模型，⁴。比较高斯-塞德尔方法的点迭代和线松弛迭代以获得更好的性能。此外，该模型在开发的Navier-Stokes求解器中实现，以验证其在仿真界面问题中的性能。将数值结果与理论解或来自可靠来源的实验数据进行比较。