Hodge decomposition of vector fields in Cartesian grids

While explicit representations of shapes such as triangular and tetrahedral meshes are often used for boundary surfaces and 3D volumes bounded by closed surfaces, implicit representations of planar regions and volumetric regions defined by level-set functions have also found widespread applications in geometric modeling and simulations. However, an important computational tool, the <i>L</i> ²-orthogonal Hodge decomposition for scalar and vector fields defined on implicit representations under commonly used Dirichlet/Neumann boundary conditions with proper correspondence to the topology presents additional challenges. For instance, the projection to the interior or boundary of the domain is not as straightforward as in the mesh-based frameworks. Thus, we introduce a comprehensive 5-component Hodge decomposition that unifies normal and tangential components in the Cartesian representation. Numerical experiments on various objects, including singlecell RNA velocity, validate the effectiveness of our approach, confirming the expected rigorous <i>L</i> ²-orthogonality and the accurate cohomology.