Mixed finite element method of a seawater intrusion problem in confined aquifers

We consider a model mixing sharp and diffuse interface approach for the seawater intrusion phenomenon in confined aquifers. The aim of this work is to introduce and analyze a new mixed formulation, obtained by writing the problem into a matrix form, and introducing a new variable σ = R(u)∇u representing the flux tensor of the primal variable u = (h, Φf)T. Here, h represents the depth of the salt/freshwater interface, Φf, the hydraulic head of freshwater, and R(u) a symmetric and positive definite diffusion matrix. We show that the continuous problem is well-posed. For the time discretization of this new mixed formulation, we use a semi-implicit scheme, and we show that the problem is well posed