On finite element methods for heterogeneous elliptic problems

Dealing with variational formulations of second order elliptic problems with discontinuous coefficients, we recall a single field minimization problem of an extended functional presented by Bevilacqua et al. [Bevilacqua, L., Feijóo, R.A., Rojas, L.F., 1974. A variational principle for the Laplace operator with application in the torsion of composite rods. International Journal of Solids Structuring 10, 1091–1102], which we associate with the basic idea supporting discontinuous Galerkin finite element methods. We review residual based stabilized mixed methods applied to Darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity. For smooth interfaces, the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of Darcy velocity on the interface. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions.