AI 中文总结
本文提出R3MG-C预条件子，通过R树划分支撑点构造聚合体，嵌入高阶局部不连续多项式空间扩展Nicolaides粗空间，作为CG预条件子在高阶有限元离散中性能优于标准AMG。
AI 中文摘要
代数多重网格（AMG）方法是针对椭圆型偏微分方程低阶离散化产生的线性方程组的鲁棒且高效的黑箱预条件子，但它们在高阶方法中的性能通常会下降。本文提出一种面向连续拉格朗日有限元离散化的代数-几何多重网格预条件子，该方法仅利用有限元支撑点的坐标自动构造延拓算子和Galerkin层级，无需预设网格层级或区域分解。支撑点通过基于轴对齐包围盒的R树算法递归划分，生成聚合体层级。与传统AMG通常使用分段常数聚合粗空间不同，本文方法通过连续节点插值嵌入定义在聚合体盒子上的p'>0阶局部不连续多项式空间，实现了Nicolaides粗空间的高阶扩展。在均匀盒正则性、插值稳定性和稳定分解假设下，两层分析量化了粗多项式阶数如何抵消大聚合体和高阶精细离散化的影响。二维和三维数值实验表明，所得V循环作为共轭梯度预条件子时，迭代计数保持稳定，在标准AMG性能下降的场景中仍有效。
英文摘要
Algebraic multigrid (AMG) methods are robust and efficient black-box preconditioners for linear systems arising from low-order discretizations of elliptic partial differential equations, but their performance often deteriorates for high-order methods. We introduce an algebraic-geometric multilevel preconditioner for continuous lagrangian finite element discretizations. The method automatically constructs prolongation operators and a Galerkin hierarchy using only the coordinates of finite element support points, without requiring a prescribed mesh hierarchy or domain decomposition. The support-point are recursively partitioned by an R-tree algorithm based on axis-aligned bounding boxes, producing a hierarchy of agglomerates. In contrast to classical AMG, which typically uses piecewise-constant aggregate coarse spaces, our method embeds local discontinuous polynomial spaces of degree $p'>0$, defined on the agglomerate boxes, through continuous nodal interpolation. This yields a high-order extension of the Nicolaides coarse space. A two-level analysis quantifies how the coarse polynomial degree offsets the effects of large agglomerates and high-order fine discretizations, under uniform box-regularity, interpolation-stability, and stable-decomposition assumptions. Numerical experiments in two and three dimensions show that the resulting V-cycle, used as a conjugate-gradient preconditioner, maintains stable iteration counts and remains effective in regimes where standard AMG deteriorates.
Comments45 pages, 5 figures, 33 tables
