R3MG: R-tree based agglomeration of polytopal grids with applications to multilevel methods

TL;DR

This paper introduces an automated, dimension-independent agglomeration method for polygonal and polyhedral grids using R-tree spatial indices, enhancing multilevel methods for PDEs with efficient hierarchy construction.

Contribution

The novel R-tree based agglomeration strategy automatically generates balanced, nested hierarchies of grid agglomerates, improving efficiency and robustness for multilevel PDE solvers.

Findings

Effective in 3D geometries with polygonal discontinuous Galerkin methods

Enables fast queries and hierarchy construction for multigrid preconditioners

Automates agglomeration process, reducing manual intervention

Abstract

We present a novel approach to perform agglomeration of polygonal and polyhedral grids based on spatial indices. Agglomeration strategies are a key ingredient in polytopal methods for PDEs as they are used to generate (hierarchies of) computational grids from an initial grid. Spatial indices are specialized data structures that significantly accelerate queries involving spatial relationships in arbitrary space dimensions. We show how the construction of the R-tree spatial database of an arbitrary fine grid offers a natural and efficient agglomeration strategy with the following characteristics: i) the process is fully automated, robust, and dimension-independent, ii) it automatically produces a balanced and nested hierarchy of agglomerates, and iii) the shape of the agglomerates is tightly close to the respective axis aligned bounding boxes. Moreover, the R-tree approach provides a full hierarchy of nested agglomerates which permits fast query and allows for efficient geometric multigrid methods to be applied also to those cases where a hierarchy of grids is not present at construction time. We present several examples based on polygonal discontinuous Galerkin methods, confirming the effectiveness of our approach in the context of challenging three-dimensional geometries and the design of geometric multigrid preconditioners.