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引用次数: 0
摘要
SIAM 科学计算期刊》,第 46 卷第 3 期,第 B179-B204 页,2024 年 6 月。 摘要这项工作描述了针对[math]中的高阶有限元问题开发的无矩阵 GPU 加速求解器。这些求解器适用于鞍点公式中的梯度二维和达西问题,并可应用于辐射扩散和多孔介质流动问题等。利用插值-组配基础(参见 [W. Pazner, T. Kolev.Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp.利用这些近似值,块预处理 MINRES 可以在与网格大小和多项式度无关的迭代次数内收敛。近似舒尔补数采用 M 矩阵图拉普拉奇的形式,因此可以通过高度可扩展的代数多网格方法进行良好预处理。针对求解算法的所有组成部分,我们开发了高性能 GPU 加速算法,并对其进行了讨论和基准测试。在一些具有挑战性的测试案例中展示了数值结果,包括 "弯曲管道 "梯度计算问题、SPE10 储层建模基准问题和非线性辐射扩散测试案例。
Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in [math]
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B179-B204, June 2024. Abstract. This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in [math]. The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation–histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675–A702]), efficient matrix-free preconditioners can be constructed for the [math]-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the “crooked pipe” grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case.
期刊介绍:
The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems.
SISC papers are classified into three categories:
1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms.
2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist.
3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.