A finite volume scheme for an elliptic equation with heterogeneous coefficients. Application to a homogenization problem Original Research Article

The aim of this paper is to develop a finite volume scheme to numerically solve an elliptic equation with heterogeneous coefficients, by using a regular mesh, independent of coefficients discontinuities. The construction of this scheme stems from a primal–dual mixed variational formulation with the solution decomposed into a regular part whose approximation belongs to a standard finite element space and an irregular part that is approximated by bubble functions. The irregular part is eliminated in terms of the regular part by imposing the continuity of the numerical flux across the lines of discontinuities of the coefficient. Several examples are presented and numerical experiments prove the efficiency of the developed method.