Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients
Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients
复制标题
DOI：
10.1016/j.cam.2005.05.019
复制
发表时间：
2006-05
影响因子：
2.4
通讯作者：
E. Flauraud;F. Nataf;F. Willien
中科院分区：
文献类型：
--
作者：
E. Flauraud;F. Nataf;F. Willien
当一个问题的系数有几个数量级的跳跃和各向异性，许多预条件和区域分解方法（DDM）遭受高原的收敛，由于存在非常小的孤立的特征值在预条件的线性系统的频谱。改进预条件子的一种方法是使用一种称为收缩的线性代数技术，或者非常类似的粗网格校正。在这两种情况下，有必要识别和计算，至少近似地，对应于“坏”特征值的所有特征向量。在DDM的框架下，我们提出了一种方法来设计接口条件，使收敛速度快，没有任何高原。该方法仅依赖于辅助矩阵的最小和最大特征值的知识。不使用特征向量。该方法依赖于货车der Sluis关于对称正定矩阵的拟最优对角预条件子的结果。这样就可以对整个接口只使用一个真实的参数来设计Robin接口条件。通过添加第二个真实的参数和更一般的界面条件，可以考虑高度非均匀和各向异性的介质。给出了数值结果，并与其它方法进行了比较.
When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the “bad” eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis’ result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches.