A face-centred finite volume approach for coupled transport phenomena and fluid flow

We present a particular derivation of the face-centred finite volume (FCFV) method and study its performance in non-linear, coupled transport problems commonly encountered in geoscientific and geotechnical applications. The FCFV method is derived from the hybridisable discontinuous Galerkin formulation, using a constant degree of approximation for the discretization of the unknowns defined on the mesh faces (edges in two dimensions). The piecewise constant degrees of freedom are determined in a global problem over the mesh skeleton. Then, the solution and its gradient are recovered at the cells centroid in a set of element-by-element independent postprocesses, both exhibiting linear convergence. The formulation of the transient advection-diffusion-reaction equation is presented in detail and the numerical analysis under challenging advective/diffusive regimes is studied. Finally, we use several numerical examples to illustrate the advantages and limitations of the FCFV method to solve problems of geoscientific and geotechnical relevance governed by the non-linear coupling between advection-diffusion-reactive transport and Stokes flow. Our results show that the FCFV method is an attractive and highly competitive alternative to other commonly used methods.