基于全近似格式的高阶间断Galerkin方法的自适应多重网格
Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme
AI总结：
该研究提出一种基于全近似格式的自适应多重网格方法，可实现六面体网格局部hp加密，经实验验证其效率、鲁棒性及动态并行网格适配能力，还初步扩展至不可压缩Navier-Stokes问题。
AI中文摘要：
针对椭圆问题的间断Galerkin格式，我们提出了一种使用Brandt全近似格式（FAS）的自适应多重网格（MG）方法。与常规方法不同，该方法可实现六面体网格的局部hp加密，无需悬挂节点。FAS-MG方法的核心组件是重叠Schwarz smoother，可选择性地用Krylov方法加速。该 smoother 专为非结构化曲线网格设计，但保留了张量积结构以实现快速对角化。数值实验表明FAS-MG方法具有极高效率，专项研究证实其对高纵横比、单元变形及不规则网格拓扑的鲁棒性。我们还使用Červený、Dobrev和Kolev（SIAM J. Sci. Comp. 41, 2019）的波前基准验证了其动态并行网格适配能力，最后给出了将该方法扩展至不可压缩Navier-Stokes问题的初步结果。
英文摘要：
We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.
