# Scalable adaptive PDE solvers in arbitrary domains

arXiv: https://arxiv.org/abs/2108.03757 · HTML: https://mlsanity.com/paper/2108.03757/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Aug 2021.

## Built on by (most cited first, 2 of 2)

- [FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometries](https://mlsanity.com/paper/2409.18032/) (2024)
- [Neural PDE Solvers for Irregular Domains](https://mlsanity.com/paper/2211.03241/) (2022)

## Abstract (arXiv)

Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a `good' adaptively-refined mesh. In this work, we present an efficient novel octree-based adaptive discretization approach capable of carving out arbitrarily shaped void regions from the parent domain: an essential requirement for fluid simulations around complex objects. Carving out objects produces an textitincomplete octree. We develop efficient top-down and bottom-up traversal methods to perform finite element computations on textitincomplete octrees. We validate the framework by (a) showing appropriate convergence analysis and (b) computing the drag coefficient for flow past a sphere for a wide range of Reynolds numbers (O(1-106)) encompassing the drag crisis regime. Finally, we deploy the framework on a realistic geometry on a current project to evaluate COVID-19 transmission risk in classrooms.