Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

TL;DR

The paper addresses robustness of smoothed aggregation algebraic multigrid for scalar PDEs with strongly heterogeneous and anisotropic materials by introducing a material-aware strength-of-connection based on a material-weighted distance Laplacian $S^{\text{dlap}}_{\sigma}$. This approach, together with tailored dropping criteria and prolongator filtering, preserves essential problem structure across material interfaces and anisotropy directions, leading to improved coarse-grid representations and convergence. Through extensive 2D/3D tests on two-domain problems, anisotropic diffusion, thermally activated batteries, and solar cells, the method demonstrates superior convergence and parallel scalability compared to classical SA-AMG and distance-based measures, with robust performance across a wide range of drop tolerances. The results suggest strong practical impact for large-scale simulations in engineering applications, and the authors provide open-source Trilinos/MueLu implementations and future directions toward Maxwell, elasticity, and multiphysics problems.

Abstract

This paper introduces a material-aware strength-of-connection measure for smoothed aggregation algebraic multigrid methods, aimed at improving robustness for scalar partial differential equations with heterogeneous and anisotropic material properties. Classical strength-of-connection measures typically rely only on matrix entries or geometric distances, which often fail to capture weak couplings across material interfaces or align with anisotropy directions, ultimately leading to poor convergence. The proposed approach directly incorporates material tensor information into the coarsening process, enabling a reliable detection of weak connections and ensuring that coarse levels preserve the true structure of the underlying problem. As a result, smooth error components are represented properly and sharp coefficient jumps or directional anisotropies are handled consistently. A wide range of academic tests and real-world applications, including thermally activated batteries and solar cells, demonstrate that the proposed method maintains robustness across material contrasts, anisotropies, and mesh variations. Scalability and parallel performance of the algebraic multigrid method highlight the suitability for large-scale, high-performance computing environments.

Figures (17)

• Figure 1: Visual representation of the error $e_k$ over the domain $\Omega$ with directional components at different $x$- and $y$-coordinates after a few steps $k$ of a Jacobi iteration.
• Figure 2: Display of the uniform, quadrilateral discretization of the test problem and the relevant neighborhood for the strength-of-connection measure.
• Figure 3: Visualization of the strength-of-connection $S$ of a node to its direct neighbors based on the material-based distance Laplacian measure for different material contrast ratios $\kappa$.
• Figure 4: Graphical illustration of the modified matrix graph $\mathcal{G}\left(C^{pw}(S_{\sigma}^{dlap})\right)$ with "unimportant" entries removed and the respective aggregates $\mathcal{A}$ build on it. Nodes associated with the Dirichlet boundary have been isolated in $\mathcal{G}\left(C^{pw}(S_{\sigma}^{dlap})\right)$ as well and will not be propagated to coarser levels. It can be clearly seen that the material information enters the aggregation process as intended.
• Figure 5: Number of iterations shown over different drop tolerances $\theta$ for combinations of dropping criteria $C^{\text{pw}}$, $C^{\text{cut-drop}}$ and strength-of-connection measures $S^{sa}$ , $S^{dlap}$ and $S_{\sigma}^{dlap}$ shown for different material contrasts $\kappa$ and mesh refinements $h$. Configurations that fail to reach the designated coarse grid size or fail to converge are not shown, as for example, if the algebraic multigrid coarsening stagnates. The material based dropping shows robustness and performance across a wide range of dropping tolerances $\theta$.