  This work proposes a novel finite volume paradigm, the face-centred finite
volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV)
centred finite volume methods, the novel FCFV defines the solution on the mesh
faces (edges in 2D) to construct locally-conservative numerical schemes.
  The idea of the FCFV method stems from a hybridisable discontinuous Galerkin
(HDG) formulation with constant degree of approximation, thus inheriting the
convergence properties of the classical HDG. The resulting FCFV features a
global problem in terms of a piecewise constant function defined on the faces
of the mesh. The solution and its gradient in each element are then recovered
by solving a set of independent element-by-element problems.
  The mathematical formulation of FCFV for Poisson and Stokes equation is
derived and numerical evidence of optimal convergence in 2D and 3D is provided.
Numerical examples are presented to illustrate the accuracy, efficiency and
robustness of the proposed methodology. The results show that, contrary to
other FV methods, the accuracy of the FCFV method is not sensitive to mesh
distortion and stretching. In addition, the FCFV method shows its better
performance, accuracy and robustness using simplicial elements, facilitating
its application to problems involving complex geometries in 3D.
