Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices
Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices
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DOI：
10.1007/s00791-015-0239-x
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发表时间：
2013-12
影响因子：
--
通讯作者：
P. Gerds;L. Grasedyck
中科院分区：
文献类型：
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作者：
P. Gerds;L. Grasedyck
我们提出了一种解决离散椭圆偏微分方程特征值问题的新方法。新方法将域分解的思想（如自动多级子结构（短 AMLS））与分层矩阵（短矩阵）的概念相结合，以获得在离散空间大小上几乎最佳缩放的求解器。尽管 AMLS 方法对于二维的偏微分方程非常有效，但在三维情况下却变得非常昂贵，因为域分解中的界面耦合需要密集矩阵运算。我们通过使用数据稀疏的分层矩阵来解决这个问题。除了离散化误差之外，我们的新方法还涉及由 AMLS 引起的投影误差和由矩阵近似引起的算术误差。在示例中研究了平衡这些误差的参数的适当选择。
We propose a new method for the solution of discretised elliptic PDE eigenvalue problems. The new method combines ideas of domain decomposition, as in theautomated multi-level substructuring(short AMLS), with the concept ofhierarchical matrices(short-matrices) in order to obtain a solver that scales almost optimal in the size of the discrete space. Whereas the AMLS method is very effective for PDEs posed in two dimensions, it is getting very expensive in the three-dimensional case, due to the fact that the interface coupling in the domain decomposition requires dense matrix operations. We resolve this problem by use of data-sparse hierarchical matrices. In addition to the discretisation error our new approach involves a projection error due to AMLS and an arithmetic error due to-matrix approximation. A suitable choice of parameters to balance these errors is investigated in examples.