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

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.