Multigrid Methods for a Class of Saddle Point Problems

在这项工作中开发和研究的快速多重网格方法有望对大量实际问题的数值解产生更广泛的影响。重要的应用包括：矢量（Hodge）拉普拉斯方程、Maxwell方程、Stokes方程、Oseen方程和Navier-Stokes方程、磁流体动力学（MHD）等。特别是MHD，在熔合技术和铸造工艺的发展中具有重要的应用。在这些应用中，由于目前不可能进行实验，因此对相应的偏微分方程进行数值模拟是必不可少的。这些模拟非常具有挑战性，需要大量的计算资源。在这个项目中开发的多网格求解器为求解越来越精确的模型提供了潜力。此外，我们在算法开发方面的改进将对许多其他领域产生影响，例如图像处理和计算机图形学。本课题分为算法开发和理论分析两部分。在算法开发方面，将开发基于有限元外部演算（FEEC）的混合有限元离散化多网格求解器。在我们的研究中，有效的平滑器是多网格方法的关键，将利用现有的有效的预处理器或分割方案来开发。其中一个例子是本项目提出的分布平滑器，它与计算流体动力学中使用的众所周知的投影方法高度相关。除了算法的发展，鞍点问题的多网格方法的更完整的收敛理论将被发展。该理论旨在打破现有工作中的强规律性假设。因此，我们的理论可以应用于更实际的问题，特别是具有奇点的解。我们的理论研究也将为算法的发展提供见解，例如，近似分布平滑和Schwarz平滑的构造，以及在几个平滑中使用的松弛参数的最佳选择。

The fast multigrid methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Important applications include: vector (Hodge) Laplacian, Maxwell equations, Stokes equations, Oseen and Navier-Stokes equations, and Magnetohydrodynamics (MHD) etc. MHD, in particular, has important applications in the development of fusion technology and casting processes. In these applications, since no experimentation is nowadays possible, the numerical simulation of the corresponding partial differential equations is indispensable. These simulations are very challenging, requiring large computational resources. The multigrid solvers developed in this project offer the potential for increasingly accurate models to be solved. In addition, our improvements in algorithm developments will have impact on many other areas, such as image processing, and computer graphics.This project is divided into two parts: algorithmic development and theoretical analysis. For the algorithmic development, multigrid solvers will be developed for mixed finite element discretization based on Finite Element Exterior Calculus (FEEC). In our study, effective smoothers, which are the key of multigrid methods, will be developed by using existing effective preconditioners or splitting schemes. One such example is a distributive smoother proposed in this project which is highly related to the well-known projection methods used in computational fluid dynamic. In addition to the algorithmic development, a more completed convergence theory of multigrid methods for saddle point problems will be developed. This theory aims to relax the strong regularity assumption in existing work. Consequently our theory can be applied to more realistic problems especially for solutions with singularities. Our theoretical investigation will also provide insight for the algorithmic development, e.g., the construction of approximated distributive smoothers and Schwarz smoothers, and optimal choice of relaxation parameters used in several smoothers.