Parallel algebraic multigrid methods on distributed memory computers for industrial CFD problems

本课题的首要目标是高效实现（顺序）ruge - st<e:1>经典AMG算法，并在实际工程应用中进行广泛测试。第二个目标是研究适用于分布式存储计算机的新的并行方法，并实现基于（无重叠）域分解思想的有限元离散化的通用并行代数多重网格算法。特别地，将研究粗化阶段和平滑算子的不同并行策略。为了提高算法的鲁棒性，采用自适应过程。虽然AMG算法适用于广泛的问题，但在某些情况下，它需要特别注意，例如，在拉伸网格上，获得鲁棒和有效的解决方案仍然是一个问题。其中一个原因是，有时，插值算子不能充分地插值误差的平滑模式。因此，将沿着基于元素的代数多重网格（AMGe）和无元素的代数多重网格（AMGe）的路线提供，分析和实现光滑误差的表征。最后，将对兼容松弛算法进行仔细的分析和（实现），因为它对并行计算有很大的希望。该项目将为博士水平的高素质人才提供重要的培训。当前项目的结果将是开发高效和健壮的并行代数多网格算法，该算法可根据机器和问题的大小进行缩放，以及可移植、模块化和基于mpi的面向对象软件。蒙特利尔的航空航天工程行业将有兴趣将（成功的）结果整合到他们的操作中，并从预期的结果中受益，即算法、并行代码和合格的专业知识。

(PLEASE READ DETAILED ABSTRACT) The first aim of the proposed project is to efficiently implement the (sequential) Ruge-Stüben classical AMG algorithm, and extensively test it on real life engineering applications. The second aim is to investigate new parallel approaches suitable for distributed memory computers, and implement a general parallel algebraic multigrid algorithm for finite element discretizations based on (non-overlapping) domain decomposition ideas. In particular, different parallel strategies for the coarsening phase and the smoothing operator will be examined. As for the robustness of the algorithm, an adaptive process will be utilized. While the AMG algorithm works well for a wide range of problems, there are situations in which it requires special care, e.g., on stretched grids where obtaining a robust and efficient solution still remains problematic. One reason for this is that, sometimes, the interpolation operator does not interpolate adequately the smooth modes of the error. Hence, a characterization of the smooth error will be provided, analyzed and implemented along the lines of element-based algebraic multigrids (AMGe), and element-free AMGe. Lastly, a careful analysis and (implementation) of the compatible relaxation algorithm, will be undertaken, as it holds much promise for parallel computation. The project will provide significant training of highly qualified personnel at the PhD level. The outcome of the current project will be the development of efficient and robust parallel algebraic multigrid algorithms which scale both with machine and problem size, and an object-oriented software that is portable, modular and MPI-based. The aerospace engineering industry in Montreal would be interested in integrating the (successful) results into their operations, and benefiting from the anticipated outcomes, namely the algorithms, the parallel code, and the qualified expertise.