Algebraic multigrid for systems of elliptic boundary‐value problems
Algebraic multigrid for systems of elliptic boundary‐value problems
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椭圆边界值问题系统的代数多重网格
DOI：
10.1002/nla.2303
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发表时间：
2021
影响因子：
4.3
通讯作者：
Lee, Barry
中科院分区：
文献类型：
--
作者：
Lee, Barry
本文发展了一种求解椭圆边值问题的代数多重网格（AMG）方法。众所周知，多重网格的椭圆方程组面临着许多挑战，不会出现大多数标量方程。这些挑战包括强互变量耦合、多维且可能较大的近零空间、解析未知的近零空间、粗自由度（CDOF）的精细选择以及网格间算子的复杂构造。在这篇文章中，我们只考虑选择的CDOF和构造的插值算子。该选择是Ruge-Stuben算法的扩展，使用节点自由度之间的新连接强度测量，即位于一个网格点的所有自由度与另一个网格点的所有自由度之间的连接强度测量。该测量基于为从基于松弛的过程导出的一组平滑测试向量生成的局部相关矩阵。有了这个措施，选择的CDOFs，然后确定在每个节点的强相关连接的数量，与鲁格-斯图本着色方案处理的选择。在选择了CDOF之后，使用自举AMG（BAMG）过程构造插值算子。我们将BAMG过程应用于平滑测试向量以获得互变量插值方案，或者应用于平滑测试向量的相似变量分量以获得内变量插值方案。此外，比较测量的相关性之间的变量内耦合与所有耦合之间的相关性，一个混合的变量内和变量间的插值方案的开发。我们进一步研究了一个间接的BAMG方法，明确使用的系统运营商的系数在构建插值权重。最后，基于弱逼近准则，我们考虑一个简单的方案来适应插值的阶数（即，调整计算域上细网格点可以插值的粗网格点的口径或最大数量）。
This article develops an algebraic multigrid (AMG) method for solving systems of elliptic boundary‐value problems. It is well known that multigrid for systems of elliptic equations faces many challenges that do not arise for most scalar equations. These challenges include strong intervariable couplings, multidimensional and possibly large near‐nullspaces, analytically unknown near‐nullspaces, delicate selection of coarse degrees of freedom (CDOFs), and complex construction of intergrid operators. In this article, we consider only the selection of CDOFs and the construction of the interpolation operator. The selection is an extension of the Ruge–Stuben algorithm using a new strength of connection measure taken between nodal degrees of freedom, that is, between all degrees of freedom located at a gridpoint to all degrees of freedom at another gridpoint. This measure is based on a local correlation matrix generated for a set of smoothed test vectors derived from a relaxation‐based procedure. With this measure, selection of the CDOFs is then determined by the number of strongly correlated connections at each node, with the selection processed by a Ruge–Stuben coloring scheme. Having selected the CDOFs, the interpolation operator is constructed using a bootstrap AMG (BAMG) procedure. We apply the BAMG procedure either over the smoothed test vectors to obtain an intervariable interpolation scheme or over the like‐variable components of the smoothed test vectors to obtain an intravariable interpolation scheme. Moreover, comparing the correlation measured between the intravariable couplings with the correlation between all couplings, a mixed intravariable and intervariable interpolation scheme is developed. We further examine an indirect BAMG method that explicitly uses the coefficients of the system operator in constructing the interpolation weights. Finally, based on a weak approximation criterion, we consider a simple scheme to adapt the order of the interpolation (i.e., adapt the caliber or maximum number of coarse‐grid points that a fine‐grid point can interpolate from) over the computational domain.
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影响因子：
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