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Balancing domain decomposition method of solving PDE for massively parallel computing
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 64, Issue 5, Pages: 50-58(2025)
- 作者机构：
中山大学航空航天学院，广东 深圳 518107
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.ZR20240178
CLC： O241.7Received：29 May 2025，
Accepted：13 June 2024，
Published Online：03 July 2025，
Published：25 September 2025
- 稿件说明：
移动端阅览
陈玉惠,黄诗杰,姚清河.面向大规模并行计算的区域平衡PDE求解方法[J].中山大学学报(自然科学版)(中英文),2025,64(05):50-58.
CHEN Yuhui,HUANG Shijie,YAO Qinghe.Balancing domain decomposition method of solving PDE for massively parallel computing[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(05):50-58.
陈玉惠,黄诗杰,姚清河.面向大规模并行计算的区域平衡PDE求解方法[J].中山大学学报(自然科学版)(中英文),2025,64(05):50-58. DOI： 10.13471/j.cnki.acta.snus.ZR20240178.
CHEN Yuhui,HUANG Shijie,YAO Qinghe.Balancing domain decomposition method of solving PDE for massively parallel computing[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(05):50-58. DOI： 10.13471/j.cnki.acta.snus.ZR20240178.
摘要
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
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、增加总自由度数量，比较不同情况下迭代次数的变化；并在
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和
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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
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 i
s 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
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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
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and
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mesh partitions. Finally， the diffusion-reaction equation is used to verify that BDD is more efficient than DDM in numerical solutions.
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references
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