Stabilized velocity post-processings for Darcy flow in heterogeneous porous media

TL;DR

This paper introduces stable finite element methods with post-processing techniques for accurately computing velocity fields in Darcy flow through heterogeneous porous media with discontinuous hydraulic conductivity, ensuring convergence and stability.

Contribution

It proposes a novel stabilization approach combining Galerkin and Least Squares residuals with interface stabilization to improve velocity accuracy in heterogeneous media.

Findings

Methods achieve expected convergence rates in heterogeneous media.

Numerical results confirm stability and accuracy of the proposed post-processing techniques.

Approaches effectively handle discontinuities in hydraulic conductivity.

Abstract

Stable and accurate finite element methods are presented for Darcy flow in heterogeneous porous media with an interface of discontinuity of the hydraulic conductivity tensor. Accurate velocity fields are computed through global or local post-processing formulations that use previous approximations of the hydraulic potential. Stability is provided by combining Galerkin and Least Squares (GLS) residuals of the governing equations with an additional stabilization on the interface that incorporates the discontinuity on the tangential component of the velocity field in a strong sense. Numerical analysis is outlined and numerical results are presented to illustrate the good performance of the proposed methods. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same orders of convergence predicted for homogeneous problem with smooth solutions, for both global and local post-processings.