  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 $\textit{incomplete}$ octree. We develop efficient top-down and
bottom-up traversal methods to perform finite element computations on
$\textit{incomplete}$ 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 ($\mathcal{O}(1-10^6)$)
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.
