基于持久单元存储的合并无矩阵几何多重网格
Coalesced Matrix-Free Geometric Multigrid on Persistent Cell-Wise Storage
AI总结：
提出用于高阶连续有限元的几何多重网格预处理器，在冗余单元存储向量上运行，避免组装全局向量。证明相关算子作用，得出单元V循环与经典方法等效及收敛性，数值实验验证其性能。
摘要：
我们提出一种用于高阶连续有限元的几何多重网格预处理器，它完全在冗余的、按单元存储的向量上运行，在层次结构的任何级别都不会形成组装后的全局向量。在这种存储范式下，传统上使自适应多重网格复杂化的机制消失了。悬挂节点约束从不组装……
英文摘要：
We present a geometric multigrid preconditioner for high-order continuous finite elements that operates entirely on redundant, cell-wise stored vectors: the assembled global vector is never formed on any level of the hierarchy. In this storage paradigm the machinery that classically complicates adaptive multigrid dissolves. Hanging-node constraints are never assembled: we prove that the plain tensor-product transfer operators, applied to the \emph{unassembled} residual, algebraically reproduce the classical constrained restriction, including the action of the transposed constraint matrix, and the edge operators of local smoothing reduce to a pointwise masking of the residual, with no splitting of the level operator into interior and edge blocks. As a consequence, the single inter-cell primitive of the whole V-cycle can use a topologically oblivious structured kernel even on adaptively refined meshes. We prove that the resulting cell-wise V-cycle is equivalent, iterate by iterate, to the classical local multigrid method, and therefore inherits its convergence theory. Numerical experiments for the Laplace operator confirm grid-independent convergence that is essentially unaffected by local refinement; on a single GPU, using nothing more than a masked point-Jacobi smoother, the solver sustains up to $1.1$\, GDoF/s per V-cycle in double precision and reaches end-to-end solve throughput on par with patch-smoother-based solvers.
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