Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

TL;DR

This paper presents a material-aware strength-of-connection measure for smoothed aggregation algebraic multigrid methods, improving robustness and convergence for heterogeneous and anisotropic materials in large-scale PDE problems.

Contribution

It introduces a novel strength-of-connection measure that incorporates material tensor information, enhancing multigrid coarsening for complex material properties.

Findings

Improved robustness across material contrasts and anisotropies.

Effective handling of sharp coefficient jumps and directional anisotropies.

Demonstrated scalability and parallel performance in real-world applications.

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.