论文标题
基于最佳局部近似空间的完全代数且健壮的两级Schwarz方法
A fully algebraic and robust two-level Schwarz method based on optimal local approximation spaces
论文作者
论文摘要
两级域分解预处理导致迭代求解器的快速收敛性和可伸缩性。但是，对于高度异质问题，系数函数在几个可能的非分离尺度上迅速变化，预先处理系统的条件数通常取决于系数函数的对比，从而导致收敛的恶化。通过合适的局部特征值问题构建的粗空增强方法，也称为自适应或光谱粗空间，恢复了稳健的，与对比度无关的收敛。但是，这些特征值问题通常依赖于非代数信息，因此无法从完全组装的系统矩阵中构建自适应粗糙空间。在本文中，提出了一个新型的代数自适应粗糙空间，该空间依赖于（局部）有限元（Fe）空间的A-实封的分解，以求解偏差方程（PDE）的函数，并提出了某些痕量和Fe函数在边界上零的迹线和Fe函数。特别是，基础是由与域分解边缘相关的两种类型的局部特征值问题的本征构成的。为了近似于局部解决PDE的函数，我们采用了转移特征值问题，该问题最初是为建造多尺度方法的最佳局部近似空间而提出的。此外，我们还利用了一个差异的特征值问题，这是对自适应广义的Dryja-Smith-Widlund（AGDSW）粗糙空间中使用的Neumann特征值问题的略微修改。两个特征值问题都完全依赖于局部的Dirichlet矩阵，这些矩阵可以从完全组装的系统矩阵中提取。通过结合多尺度和域分解方法的参数，我们得出了与对比型的上限有关条件数的界限。
Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems, where the coefficient function is varying rapidly on several possibly non-separated scales, the condition number of the preconditioned system generally depends on the contrast of the coefficient function leading to a deterioration of convergence. Enhancing the methods by coarse spaces constructed from suitable local eigenvalue problems, also denoted as adaptive or spectral coarse spaces, restores robust, contrast-independent convergence. However, these eigenvalue problems typically rely on non-algebraic information, such that the adaptive coarse spaces cannot be constructed from the fully assembled system matrix. In this paper, a novel algebraic adaptive coarse space, which relies on the a-orthogonal decomposition of (local) finite element (FE) spaces into functions that solve the partial differential equation (PDE) with some trace and FE functions that are zero on the boundary, is proposed. In particular, the basis is constructed from eigenmodes of two types of local eigenvalue problems associated with the edges of the domain decomposition. To approximate functions that solve the PDE locally, we employ a transfer eigenvalue problem, which has originally been proposed for the construction of optimal local approximation spaces for multiscale methods. In addition, we make use of a Dirichlet eigenvalue problem that is a slight modification of the Neumann eigenvalue problem used in the adaptive generalized Dryja-Smith-Widlund (AGDSW) coarse space. Both eigenvalue problems rely solely on local Dirichlet matrices, which can be extracted from the fully assembled system matrix. By combining arguments from multiscale and domain decomposition methods we derive a contrast-independent upper bound for the condition number.
