APPROXIMATION BY THE FINITE VOLUME METHOD OF AN ELLIPTIC-PARABOLIC EQUATION ARISING IN ENVIRONMENTAL STUDIES

We prove the convergence of a finite volume scheme for the Richards equation β(p)t-div(Λ(β(p))(∇p-ρg))=0 together with a Dirichlet boundary condition and an initial condition in a bounded domain Ω ×(0, T). We consider the hydraulic charge $u = \frac{p}{\rho g} - z$ as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L2 of the water saturation β(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov's theorem. This implies the convergence in L2 of the approximate water mobility towards Λ(β(p)) as the time and mesh steps tend to 0, which in turn implies the convergence of the approximate pressure to a weak solution p of the continuous problem.