A Conforming Primal-Dual Mixed Formulation for the 2D Multiscale Porous Media Flow Problem

Abstract: In this paper a new primal-dual mixed finite element method is introduced, aimed to model multiscale problems with several geometric subregions in the domain of interest. In each of these regions porous media fluid flow takes place, but governed by physical parameters at a different scale; additionally, a fluid exchange through contact interfaces occurs between neighboring regions. The well-posedness of the primal-dual mixed finite element formulation on bounded simply connected polygonal domains of the plane is presented. Next, the convergence of the discrete solution to the exact solution of the problem is discussed, together with the convergence rate analysis. Finally, the numerical examples illustrate the method's capabilities to handle multiscale problems and interface discontinuities as well as experimental rates of convergence.