A face-centred finite volume method for second-order elliptic problems

This work proposes a novel finite volume paradigm, ie, the face-centred finite volume (FCFV) method. Contrary to the popular vertex and cell-centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. 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 two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume 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 three dimensions.