Novel Two-Level Convergence Theory for Algebraic Multigrid Solver with Overlapping Smoothers

Researchers have developed a comprehensive convergence theory for the least-squares algebraic-multigrid domain-decomposition (LS-AMG-DD) solver, an advanced computational method designed for solving large sparse systems of linear equations. The LS-AMG-DD solver operates as an algebraic multilevel technique specifically tailored for symmetric positive definite matrices that can be expressed in Gram form A=G^T G, a structure common in finite-element discretizations across multiple problem classes.

The new two-level convergence theory demonstrates that the solver's coarse space satisfies a weak approximation property within a norm induced by an aggregate-wise block-Jacobi smoother. Crucially, the approximation constant remains bounded by a user-controlled local spectral cutoff threshold, allowing practitioners to tune solver performance. The theoretical framework combines this approximation property with established multiplicative two-level cycle theory, yielding convergence bounds that are cleanly factored by both the cutoff threshold and a smoother norm-comparison constant.

The research provides explicit bounds for the norm-comparison constant applicable to both block Jacobi and overlapping additive Schwarz smoothers. Additionally, the authors introduce a novel convergence bound for additive Schwarz methods expressed through a trivially computable constant bounded above by the coloring constant. These theoretical advances offer practitioners concrete, computable measures for predicting solver behavior.

Extensive numerical experiments validate the theoretical predictions across diverse problem types, including scalar H¹, vector H(div), and vector H(curl) finite-element problems. The experiments provide empirical evidence that the solver exhibits insensitivity to mesh refinement and polynomial degree variations, desirable properties for robust computational methods. This work advances the understanding of multilevel solver behavior and provides tools for analyzing and optimizing algebraic multigrid methods in practical applications.