Compact cell-based Compatible Discrete Operator diffusion scheme on Cartesian AMR mesh

In this paper we investigate cell-based Compatible Discrete Operator (CDO) scheme for elliptic problems. These mimetic schemes rely on mixed potential degrees of freedom at the cells and flux degrees of freedom at the faces. In this paper, we propose a compact rewriting of this scheme by eliminating a large part of the degrees of freedom associated with the faces, which greatly reduces the size of the linear systems to be inverted and therefore the computational cost. For this, we take advantage of Cartesian AMR (Adaptive Mesh Refinement) meshes, for which the discrete CDO Hodge operator can be made diagonal almost everywhere. We also propose a formulation of general Robin-type boundary conditions for these cell-based CDO schemes, also valid in the presence of Immersed Boundaries with a cut-cell approach. The proposed scheme and its performances is validated through various test cases.