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.

翻译：我们提出了一种基于空间索引的多边形与多面体网格聚合新方法。聚合策略是偏微分方程多面体方法中的关键要素，用于从初始网格生成计算网格（层级结构）。空间索引是专门的数据结构，可显著加速任意空间维度中涉及空间关系的查询。我们展示了如何通过构建任意细网格的R树空间数据库，提供一种自然且高效的聚合策略，该策略具有以下特征：（i）过程全自动化、鲁棒且维度无关；（ii）自动生成平衡且嵌套的聚合体层级；（iii）聚合体的形状紧密贴合其对应的轴向包围盒。此外，R树方法提供了完整的嵌套聚合体层级结构，支持快速查询，并使得高效的几何多重网格方法能够应用于构建时未预设网格层级的场景。我们基于多边形间断伽辽金方法展示了多个算例，验证了该方法在复杂三维几何问题与几何多重网格预条件器设计中的有效性。