A locally mass-conservative enriched Petrov-Galerkin method without penalty for the Darcy flow in porous media

TL;DR

This paper introduces a novel enriched Petrov-Galerkin method for Darcy flow in porous media that is locally mass-conservative, avoids penalty terms, and offers computational efficiency and optimal convergence.

Contribution

The paper develops a penalty-free, locally mass-conservative EPG method with a decoupled pressure solution framework, improving efficiency over existing methods.

Findings

The EPG method is locally mass-conservative.

It achieves optimal convergence rates.

It requires fewer degrees of freedom than DG methods.

Abstract

In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed method.