In this work we investigate the stability and approximation properties of the cell-vertex finite volume method applied to an elliptic partial differential equation discretized on quadrilateral or cuboid meshes in two or three dimensions respectively. The Helmholtz type equation of interest originates from the projection step in the semi-discretisation of a second order semi-implicit finite volume scheme, which is capable of resolving the pseudo-incompressible and compressible regime of the Euler equations in a unified numerical framework.
Consequently, we investigate the mixed saddle point problem determined by the pseudo-incompressible divergence constraint and include the source terms responsible for compressible effects. We provide stability and an a-priori error estimate for the projection step in the pseudo-incompressible case, as well as stability for the compressible situation. To this end we leverage an interpretation of the discrete flux variables in terms of discontinuous Galerkin method and introduce the Raviart-Thomas interpolation operator on the dual control volumes surrounding each vertex of the primary grid. This choice is motivated by the natural divergence defined via the integral normal flux passing through the boundary of a dual control volume.