A black-box, multilevel algebraic preconditioning framework for conforming finite elements

Abstract

Recently we introduced the least-squares algebraic-multigrid domain-decomposition (LS-AMG-DD) method as a multilevel, algebraic preconditioner for sparse symmetric positive definite (SPD) matrices that admit a Gram representation \(A=G^{\top}G\) \cite{southworth2026lsamgdd}. The factor \(G\) induces a local symmetric positive semidefinite (SPSD) splitting of \(A\) used to define local spectral problems from which an interpolation $P$ is built, and a coarse-level Gram operator induced under Galerkin coarsening, \(A_c=G_c^\top G_c\), for \(G_c:=GP\). This paper clarifies when this Gram structure arises, showing that, on a prescribed degree-of-freedom cover \({\cal C}\), a \({\cal C}\)-local Gram representation of $A$ exists if and only if \(A\) admits a \({\cal C}\)-local SPSD splitting. We then connect this viewpoint to conforming finite-element discretizations, where bilinear forms are naturally assembled from elementwise SPSD energies and therefore admit element-local Gram representations after choosing local factors (e.g., via algebraic factorizations of element blocks). Taken together, these observations provide an essentially black-box route for applying LS-AMG-DD to conforming finite-element problems. Numerical tests illustrate the robustness of the method on several problems for which classical AMG methods require more than $10^5$ iterations to converge, including high-order discretizations of grad--div in \(\hdiv\), anisotropic hyperdiffusion in $H^2$, and linear elasticity in vector \(H^1\). Moreover, in some comparisons with existing AMG methods, LS-AMG-DD produces errors that are 2--5 orders of magnitude smaller, even when all methods are stopped at the same relative residual tolerance.

Disclosure

“d with all modes. Future work will focus on complexity control, because the same spectral enrichment that gives robustness can also produce large coarse spaces and significant fill-in on coarser grids. Acknowledgments. The authors used OpenAI’s ChatGPT during the prepa- ration of this manuscript, including for exploratory mathematical discussion, coding,”

PDF page 22

Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-24T00:34:15.228Z

Structural counts