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.

中文翻译：

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椭圆边值问题系统的代数多重网格

本文开发了一种代数多重网格（AMG）方法，用于解决椭圆形边值问题的系统。众所周知，椭圆方程系统的多重网格面临许多标量方程所没有的挑战。这些挑战包括强互变量耦合，多维和可能较大的近零空间，分析上未知的近零空间，精细自由度（CDOF）的精细选择以及网间算子的复杂构造。在本文中，我们仅考虑CDOF的选择和内插运算符的构造。该选择是Ruge–Stuben算法的扩展，它使用了在节点自由度之间采用的新的连接强度度量，也就是说，位于某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。此度量基于为从基于松弛的过程派生的一组平滑测试向量生成的局部相关矩阵。通过这种措施，然后根据每个节点上强关联的连接数确定CDOF，并通过Ruge–Stuben着色方案处理选择。选择了CDOF之后，使用自举AMG（BAMG）过程构造内插运算符。我们对平滑的测试向量应用BAMG程序以获得可变的插值方案，或者对平滑的测试向量的类似变量分量应用BAMG以获得变量内插值的方案。而且，通过比较变量内耦合之间的相关性和所有耦合之间的相关性，开发了混合的变量内和变量间插值方案。我们进一步研究了间接BAMG方法，该方法在构造插值权重时显式使用系统算子的系数。最后，基于弱近似准则，我们考虑一种在计算域上适应插值顺序（即，适应细网格点可以从中插值的粗网格点的口径或最大数量）的简单方案。