CONVERGENCE OF A FINITE ELEMENT-FINITE VOLUME SCHEME FOR COUPLED ELLIPTIC PARABOLIC EQUATIONS

Convergence of a finite element-finite volume scheme for coupled elliptic-parabolic equations. We study here a discretization scheme for the following coupled system of equations: [GRAPHICS] defined over a bounded open set OMEGA of R2 or R3. A triangular mesh is used for the space discretization of both equations, in the case of two space dimensions (a tetrahedral mesh is used in the 3D case). The time discretization of the first equation is performed with the explicit Euler scheme, while its space discretization uses a weighted finite volume method. A finite element method is used for the elliptic equation. The numerical scheme thus defined converges, under a usual stability condition, in the sense that a family of approximate solutions converges, when the mesh size tends to 0, towards a solution of the original coupled system. This result is proven via an estimate on the total variation of the approximate solutions. Note that it also yields the existence of a solution to this system of equations.