几类具有低计算复杂性的非重叠 DDM 预条件子及其并行实现

Krylov subspace iterative methods with preconditioners based on the nonoverlapping domain decomposition method (DDM) are popular for solving discrete systems of partial differential equations. The parallel technologies based on MPI, OpenMp, and CUDA(GPU) take full advantage of the modern computing capability of heterogeneous computers and have become important in scientific computing. In this project, we plan to study efficient nonoverlapping DDM preconditioners with lower computational complexity and better applicability. In particular, we will focus on efficient nonoverlapping DDM preconditioners for the discrete systems of some complicate electromagnetic field equations. Furthermore, we will develop efficient parallel Krylov subspace iterative solvers with these new DDM preconditioners on heterogeneous computer system. To achieve our goal, we notice that there are many difficulties and challenges, such as how to construct the reasonable coarse subspace and establish a new stable space decomposition on an unstructured grid for complicated problems; and how to develop parallel solvers with good scalability aiming at the feature of heterogeneous computers. To resolve these difficulties, we need to develop new methodologies, theories, and techniques. The achievements of this project will be of great importance in theory and practice.

基于非重叠 DDM 预条件子的 Krylov子空间迭代法是目前国际上求解偏微分方程离散化系统的流行方法之一。MPI/OpenMP+CUDA(GPU)程序实现模式已成为能充分发挥异构计算机计算能力的重要途径。本课题将重点研究具有更低计算复杂性和更好普适性的高效非重叠DDM 预条件子，同时针对几类复杂电磁场方程组的棱有限元离散系统，研究其基于非重叠DDM的高效预条件子。进一步，针对异构计算机体系，研究基于这些预条件子的Krylov子空间迭代法的高效并行解法器。与已有工作相比，我们将面临许多困难点，例如，如何针对复杂问题的非结构网格，构造合理的粗空间，给出新的稳定性空间分解；如何针对异构计算机体系的特点，研制相应的具有良好并行可扩展性的解法器等。解决这些困难，需要发展许多新的方法、理论和技术。这些研究成果具有重要的理论意义和实际应用价值。

本课题组通过项目的研究，一方面，重点针对几类复杂偏微分方程组，如线弹性方程组、Stokes方程组、大规模油藏数值模拟、惯性约束聚变等问题的离散系统，围绕具有低计算复杂性的非重叠区域分解法、多层网格法等预条件快速算法开展研究，获得了一些重要的理论和数值结果；另一方面，针对复杂几何区域的负载平衡分解、几何多层网格法和代数多层网格法等在异构体系下的高效并行实现开展研究，获得了很好的并行加速效果。此外，我们还对大规模散体颗粒系统的并行数值模拟、磁流体等问题做了一些扩展性研究。以上研究成果对丰富和发展非重叠区域分解法和异构体系下的并行程序设计具有重要的理论意义与实际应用价值。