Two-level convergence of Algebraic Multigrid with Overlapping Smoothers and Spectral Coarse Grids

Abstract

We recently developed the least-squares algebraic-multigrid domain-decomposition (LS-AMG-DD) solver as an algebraic multilevel method for sparse symmetric positive definite matrices that admit a Gram representation \(A=G^{\top}G\) \cite{southworth2026lsamgdd}. Many problem classes admit such structure, including many conforming finite-element discretizations. The solver constructs coarse spaces from local eigenproblems on nonoverlapping, algebraic aggregates and uses Schwarz-type smoothers on the induced overlapping subdomains. This paper develops a novel two-level convergence theory for this solver. Our theory shows that the solver's coarse space satisfies a weak approximation property in a norm induced by an aggregate-wise block-Jacobi smoother, and moreover, that the corresponding approximation constant is bounded by a user-controlled local spectral cutoff threshold. We combine this approximation property with standard sharp theory for multiplicative two-level cycles. The resulting two-level bound is cleanly factored by the cutoff threshold and a smoother norm-comparison constant; we derive explicit bounds for this constant for block Jacobi and overlapping additive Schwarz smoothers. We also develop a new convergence bound for additive Schwarz methods in terms of a trivially computable constant that is bounded above by the coloring constant. Numerical experiments on scalar \(H^1\), vector \(H(\operatorname{div})\), and vector \(H(\operatorname{curl})\) finite-element problems provide supporting evidence for the theory, including evidence for the solver's insensitivity to mesh refinement and polynomial degree.