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A Pattern-Based Virtual Element Method on Quadtree Meshes
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AI
DOI：10.1007/s10915-026-03380-y.png)
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Abstract
En 中文
We introduce a Pattern-Based Virtual Element Method (PBVEM) for the efficient discretization of partial differential equations on Cartesian quadtree meshes. While quadtree grids are natural for image-derived geometries, embedded-interface problems, and adaptive simulations, their hierarchical and nonconforming structure leads to significant computational overhead in standard VEM due to repeated local projection and stabilization procedures. PBVEM alleviates this cost by exploiting the finite set of admissible element topologies induced by the 2:1 balancing rule. Elements are classified into a small number of canonical patterns, for which the corresponding VEM operators are precomputed on reference configurations. Local stiffness matrices and load vectors are then assembled via inexpensive pattern lookups and scaling, avoiding redundant reconstructions and simplifying the treatment of hanging nodes without altering the underlying VEM formulation. Numerical experiments, including image-based geometries, demonstrate that PBVEM preserves optimal convergence rates while achieving substantial reductions in assembly time compared with the conventional VEM. The proposed approach provides a scalable and robust framework for large-scale, adaptive, and data-driven simulations on quadtree meshes.
Keywords:
Virtual element method
Quadtree decomposition
Image-based analysis
Adaptive refinement
Patterns
Journal
IF:
3.3
Papers:
672
Citations:
9.6K