Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

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.

💡 Research Summary

This paper addresses a long‑standing difficulty in applying algebraic multigrid (AMG), specifically smoothed‑aggregation (SA) AMG, to scalar elliptic problems that involve highly heterogeneous and anisotropic material coefficients. Classical strength‑of‑connection (SoC) measures, which are based solely on matrix entries or on simple geometric distances, fail to recognize weak couplings across material interfaces or to align with the principal directions of anisotropy. Consequently, the coarse‑grid operators generated by standard AMG do not reflect the true physics of the underlying problem, leading to poor convergence or even divergence when material contrasts span several orders of magnitude.

The authors propose a “material‑aware” SoC that incorporates the material tensor σ(x) directly into the coarsening process. The key idea is to replace the Euclidean distance used in the distance‑Laplacian SoC with a metric derived from σ. For two degrees of freedom i and j, the distance is defined as
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