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 beta(p)t - div(Delta(beta(p))(delp - rhog)) = 0 together with a Dirichlet boundary condition and an initial condition in a bounded domain Omega x (0, T). We consider the hydraulic charge u = -p/rhog - z as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L-2 of the water saturation beta(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov's theorem. This implies the convergence in L-2 of the approximate water mobility towards Lambda(beta(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.