Abstract. The Marker-and-Cell (MAC) discretization for Navier-Stokes equations is famously efficient and super-convergent for uniform grids. Practitioners of finite difference, finite volume, discontinuous Galerkin, and mixed-finite element methods all have ways of deriving the MAC scheme within their favored frameworks. But while these derivations coincide on a uniform grid, they diverge into different methods in the presence of the coarse-fine boundaries that occur in adaptively refined meshes, each with their own merits.
Inspired by local codifferential discretizations in other work (which have appeared in the multipoint flux mixed finite element method of Wheeler & Yotov and the more recent augumented serendipity elements of Lee & Winther), we show how the MAC scheme can be framed as a mixed discretization of the vector Lapacian in which the codifferential is locally eliminated. From this starting point, we can generalize to AMR meshes: quadtrees and octrees, but beyond that to any hierarchically refined mesh. The generalization retains the formal order of accuracy and sparsity of the original and respects the Helmholtz decomposition. We will also discuss the finite difference stencils generated by this approach, and the possibilities of combining this approach with higher order methods.
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