论文标题
椭圆运算符的多个和聚类特征值的两级块预处理雅各比·戴维森方法
A Two-Level Block Preconditioned Jacobi-Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators
论文作者
论文摘要
在本文中，我们提出了一个两级块预处理的Jacobi-Davidson（BPJD）方法，用于有效地解决因有限元近似值$ 2000 $ TH（$ M = 1，2 $）订单对称eLLIPTIC ELLIPTRIC ELLIPTIC EIGENVALUE问题而导致的离散特征问题。我们的方法有效地计算了前几个特征，包括具有相应特征功能的多个和聚类的特征值，尤其是。通过使用重叠域分解（DD）构建新的有效的预处理程序，该方法是高度平行的。它仅需要计算几个小规模的平行子问题，而每个迭代中的特征值问题很小。我们的理论分析表明，该方法的收敛速率由$ c（h）（1-c \ frac {δ^{2m-1}}} {h^{2m-1}}}^{2} $，其中$ h $是子域的直径，$δ$是$δ$的大小。常数$ c $与目标特征值之间的网格尺寸$ h $和内部差距无关，这表明我们的方法是最佳且群集鲁棒的。同时，$ h $依赖性常数$ c（h）$单调降低至$ 1 $，为$ h \ f \ 0 $，这意味着更多的子域会导致更好的收敛率。给出了支持我们理论的数值结果。
In this paper, we propose a two-level block preconditioned Jacobi-Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of $2m$th ($m = 1, 2$) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several eigenpairs, including both multiple and clustered eigenvalues with corresponding eigenfunctions, particularly. The method is highly parallelizable by constructing a new and efficient preconditioner using an overlapping domain decomposition (DD). It only requires computing a couple of small scale parallel subproblems and a quite small scale eigenvalue problem per iteration. Our theoretical analysis reveals that the convergence rate of the method is bounded by $c(H)(1-C\frac{δ^{2m-1}}{H^{2m-1}})^{2}$, where $H$ is the diameter of subdomains and $δ$ is the overlapping size among subdomains. The constant $C$ is independent of the mesh size $h$ and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the $H$-dependent constant $c(H)$ decreases monotonically to $1$, as $H \to 0$, which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.
