The eigenvalue problem for the 2D Laplacian in H-matrix arithmetic and application to the heat and wave equation
The eigenvalue problem for the 2D Laplacian in H-matrix arithmetic and application to the heat and wave equation
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DOI：
10.1007/s00607-003-0061-z
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发表时间：
2004-01-01
期刊：
影响因子：
3.7
通讯作者：
Lintner, M
中科院分区：
文献类型：
--
作者：
Lintner, M
Hackbusch最近引入了一类矩阵（H-矩阵），用于近似FEM和BEM应用中产生的大型和完全填充矩阵。这些矩阵是数据稀疏的，并且允许几乎线性复杂度的近似矩阵运算。在本文中，我们选择了一类特殊的H-矩阵，提供了一个很好的近似逆的离散二维拉普拉斯。对于这些二维H-矩阵，我们研究了块三角线性方程组的块递归计划和Cholesky和LDLT分解的近似算法的几乎线性复杂性。使用LDLT分解，我们计算H-矩阵算术中的离散2D拉普拉斯算子的特征对，通过所谓的同时迭代来计算由于Stewart的非Hermitian矩阵的不变子空间。我们应用H-矩阵技术近似的解决方案的高频二维波动方程的光滑的初始数据和二维热方程的任意初始数据的离散二维拉普拉斯的谱分解，对数因子，最佳的复杂性。
A class of matrices (H-matrices) has recently been introduced by Hackbusch for approximating large and fully populated matrices arising from FEM and BEM applications. These matrices are data-sparse and allow approximate matrix operations of almost linear complexity. In the present paper, we choose a special class of H-matrices that provides a good approximation to the inverse of the discrete 2D Laplacian. For these 2D H-matrices we study the blockwise recursive schemes for block triangular linear systems of equations and the Cholesky and LDLT factorization in an approximate arithmetic of almost linear complexity. Using the LDLT factorization we compute eigenpairs of the discrete 2D Laplacian in H-matrix arithmetic by means of a so-called simultaneous iteration for computing invariant subspaces of non-Hermitian matrices due to Stewart. We apply the H-matrix techniques to approximate the solutions of the high-frequency 2D wave equation for smooth initial data and the 2D heat equation for arbitrary initial data by spectral decomposition of the discrete 2D Laplacian in, up to logarithmic factors, optimal complexity.