Single-grid Multi-level Solvers for Coupled PDE Systems

该项目的目标是开发和研究一类特殊的多层方法，它结合了几何多层网格（GMG）和代数多层网格（AMG）方法的技术，我们称之为“单网格多层方法”（SGML）。重点是离散的偏微分方程，用户通常可以获得有关底层几何网格的详细信息。研究小组正在设计求解器，它使用来自最精细网格的信息（因此称为单网格方法）来选择一个简单而固定的粗化，从而允许对整个网格和多层求解器的操作复杂性进行显式控制。我们正在研究的核心新思想是，当用作平滑器时，MG松弛方案的自适应构造算法的设计和分析。与现有的AMG方法不同，其中平滑是固定的，粗化是设置阶段的关键组件，SGML将在设置阶段构建平滑，以补充其简单的基于几何的粗化过程。应该注意的是，平滑的代数构造也可以受益于使用几何网格的特性，例如，获得并行的未知数的适当划分。SGML方法（连同许多在过去十年中开发的用于构造MG插值的有前途的代数技术）也在考虑之中。然而，PI和co-PI关注的是SGML方法，因为它具有显式控制复杂性的能力，这反过来又允许（几乎）最优负载平衡和可预测的通信模式，因此该方法非常适合并行计算。总的来说，开发中的迭代求解器被设计为在开放源代码并行代码中实现，并可供科学计算社区使用。这将为未来相关领域的算法研究和开发提供一个计算框架，以及强大的仿真工具。总之，所提出的方法利用所有可用的信息构建求解器，以提高并行多核计算架构上物理现象的数值建模和模拟效率。教育活动包括培养研究生。

The goal of this project is to develop and study a special class of multilevel methods that combine techniques from the Geometric Multigrid (GMG) and Algebraic Multigrid (AMG) methodologies, which we refer to as the "single-grid multilevel method" (SGML). The focus is discretized partial differential equations, for which detailed information on the underlying geometric grid is generally available to the user. The research team is designing solvers that use information from the finest grid (hence termed the single-grid method) to select a simple and fixed coarsening that allows for explicit control of the overall grid and operator complexities of the multilevel solver. The central new idea that we are investigating concerns the design and analysis of algorithms for adaptive construction of the MG relaxation scheme when used as a smoother. In contrast to existing AMG methods, in which the smoother is fixed and coarsening is the key component in the setup phase, SGML will construct the smoother in the setup phase to complement its simple geometry-based coarsening process. It should be noted that the algebraic construction of the smoother can also benefit from using properties of the geometric grid, for example, to obtain a suitable partitioning of the unknowns in parallel. The SGML approach (together with the many of the promising algebraic techniques for constructing the MG interpolations developed over the last decade) is also under consideration. The PI and co-PIs, though, are focusing on the SGML method because of its ability to explicitly control complexity, which in turn allows for (nearly) optimal load balancing and predictable communication patterns, such that the method is well suited for parallel computing. Overall, the iterative solvers under development are designed to be implemented in open source parallel codes and made available to the scientific computing community. This will provide a computational framework for future algorithm research and development in related areas as well as powerful tools for simulation. In summary, the proposed methodology constructs solvers using all the information available to increase the efficiency of numerical modeling and simulation of physical phenomena on parallel multi-core computing architectures. Educational activities include the training of graduate students.