论文标题
重新审视具有高对比度系数的椭圆问题的域分解方法
Domain Decomposition Methods for Elliptic Problems with High Contrast Coefficients Revisited
论文作者
论文摘要
在本文中，我们重新审视了未重叠的域分解方法，以解决具有高对比度系数的椭圆问题。发现了一些有趣的结果。我们发现Dirichlet-Neumann算法和Robin-Robin算法可能充分利用系数的比率。实际上，在两个子域的情况下，我们表明它们的收敛速率为$ O（ε）$，如果$ν_1\llν_2$，其中$ε=ν_1/ν_2$和$ν_1，nν_2$是两个子域的系数。此外，对于许多子域而言，dirichlet-neumann算法和罗宾 - 罗宾算法的条件数为$ 1+ε（1+ \ log（h/h））^2 $和$ c+ε（1+\ log（h/h））^2 $分别是$ hiy $ε$，分别是$ε$的coeff coeff coeff coeff coeff。此外，Neumann-Neumann算法和Dirichlet-Dirichlet算法的收敛行为可能独立于系数，而它们无法从不连续系数中受益。数值实验是预先形成的，以确认我们的理论发现。
In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients. Actually, in the case of two subdomains, we show that their convergence rates are $O(ε)$, if $ν_1\llν_2$, where $ε= ν_1/ν_2$ and $ν_1,ν_2$ are coefficients of two subdomains. Moreover, in the case of many subdomains, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+ε(1+\log(H/h))^2$ and $C+ε(1+\log(H/h))^2$, respectively, where $ε$ may be a very small number in the high contrast coefficients case. Besides, the convergence behaviours of the Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm may be independent of coefficients while they could not benefit from the discontinuous coefficients. Numerical experiments are preformed to confirm our theoretical findings.
