Treffer: 面向大规模并行计算的区域平衡PDE求解方法.

This study investigates the numerical solution efficiency and memory consumption of Poisson equation, heat conduction equation, and wave equation, using a non-overlapping domain decomposition method(DDM). To address the large-scale and singular nature of interface problems between subdomains generated by DDM, the balanced domain decomposition(BDD)method was employed. This method integrates conjugate gradient iteration with preconditioning techniques. The parallel algorithm is based on a symmetric multiprocessing(SMP)architecture, where all processor units are equal in status and share memory. First, the implementation of DDM and BDD based on the Poisson equation is introduced. Next, the finite element discretization processes and corresponding discrete matrix forms for the three PDEs are presented. Then, by increasing the total degrees of freedom while maintaining H/h ratio, the variation in iteration counts under different conditions is compared. Additionally, the iterative efficiency and memory consumption of DDM and BDD when solving these three PDEs are analyzed and contrasted under 1 000 × 1 000 and 2 000 × 2 000 mesh partitions. Finally, the diffusion-reaction equation is used to verify that BDD is more efficient than DDM in numerical solutions. [ABSTRACT FROM AUTHOR]

利用非重叠的区域分解方法(DDM)探讨了以Poisson 方程、热传导方程和波动方程为代表的椭圆型、抛物 型和双曲型偏微分方程(PDEs)的数值求解效率及内存消耗。针对由DDM产生的子区域间界面问题规模较大且奇 异的特点, 采用了平衡区域分解(BDD)方法, 该方法结合了共轭梯度迭代法与预处理技术。所采用的并行算法基 于对称多处理器(SMP)结构, 所有处理器单元地位平等且共享内存。首先, 介绍了基于Poisson 方程的DDM和 BDD实现方法。其次, 阐述了3 种PDEs 的有限元离散过程及其对应的离散矩阵形式。然后, 通过固定H/h、增加总 自由度数量, 比较不同情况下迭代次数的变化;并在1 000 × 1 000和2 000 × 2 000剖分下, 分析了DDM和BDD在求 解这3 类PDEs 时的迭代效率与内存消耗量。最后, 通过扩散反应方程验证了BDD相较于DDM在数值求解方面具 有更高的效率. [ABSTRACT FROM AUTHOR]

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