PDE系统的节点AMG粗化与插值
Nodal AMG Coarsening and Interpolation for PDE Systems
AI总结：
本文提出一种适用于具有大近核的PDE系统（如H(curl)和H(div)）的代数多重网格（AMG）粗化算法与插值算子，基于兼容松弛和理想插值模型，通过节点对偶问题粗化与匹配算法完全解耦变量以增强稀疏性，结合自动平滑器构建了完整的双网格方法，并在Stokes等系统上验证了有效性。
摘要：
我们提出了一种为代数多重网格（AMG）方法构建实用粗化算法和插值算子的方法，专为具有大近核的偏微分方程（PDE）系统（如H(curl)和H(div)）量身定制。我们的方法基于广义AMG（GAMG）框架中的兼容松弛（CR）和理想插值模型，但引入了几项修改，以定义适用于PDE系统的AMG方法。我们通过一个粗化过程构建插值算子，该过程首先粗化一个节点对偶问题，然后使用匹配算法构建粗变量和细变量。我们的插值遵循理想公式；然而，我们通过完全解耦细变量和粗变量来增强理想插值的稀疏性。当粗变量与几何细化对齐时，我们的方法在非结构网格上重现重新离散化。结合一种识别局部近核的自动平滑器构建方案，我们的方法形成了一个完整的双网格方法。最后，我们还展示了数值结果，通过将其应用于目标问题和Stokes系统，证明了该插值方案的有效性。
英文摘要：
We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) and H(div). Our method builds on compatible relaxation (CR) and the ideal interpolation model within the generalized AMG (GAMG) framework but introduces several modifications to define an AMG method for PDE systems. We construct an interpolation operator through a coarsening process that first coarsens a nodal dual problem and then builds the coarse and fine variables using a matching algorithm. Our interpolation follows the ideal formulation; however, we enhance the sparsity of ideal interpolation by decoupling the fine and coarse variables completely. When the coarse variables align with the geometric refinement, our method reproduces re-discretization on unstructured meshes. Together with an automatic smoother construction scheme that identifies the local near kernels, our approach forms a complete two-grid method. Finally, we also show numerical results that demonstrate the effectiveness of this interpolation scheme by applying it to targeted problems and the Stokes system.
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