Convergence analysis of a Finite Difference method for 2D-flow problems with a uniform full permeability tensor

We present in this work a convergence analysis of a Finite Difference method for solving on quadrilateral meshes $\mathrm{2D}$-flow problems in homogeneous porous media with a full permeability tensor. We start with the derivation of the discrete problem by using our finite difference formula for a mixed derivative of second order. A result of existence and uniqueness of the solution for that problem is given via the positive definiteness of its associated matrix. Their theoretical properties, namely, stability on the one hand (with the associated discrete energy norm) and error estimates (with $L^2$-norm, relative $L^2$-norm and $L^\infty$-norm ) are investigated. Numerical simulations are shown.