On Finite Element Methods for Heterogeneous Elliptic Problems

TL;DR

This paper reviews finite element methods for second order elliptic problems with discontinuous coefficients, focusing on stabilized mixed methods for heterogeneous media and convergence properties.

Contribution

It extends residual based stabilized mixed methods to heterogeneous media with interfaces, preserving flux continuity and interface constraints.

Findings

Convergence rates match those of homogeneous problems with smooth solutions.

Formulations effectively handle discontinuities in coefficients and interfaces.

Preservation of flux continuity across interfaces in heterogeneous media.

Abstract

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 (1974), 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.