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

Huangxin Chen, Piaopiao Dong, Shuyu Sun, Zixuan Wang

TL;DR

The paper tackles the lack of local mass conservation in standard continuous Galerkin methods for Darcy flow in porous media by introducing an enriched Petrov-Galerkin (EPG) method. The EPG framework enriches the CG trial space with bubble functions and the test space with piecewise constants, enabling a penalty-free formulation and a decoupled two-step pressure solve: first compute $p_h^c$ in $M_0^k(\mathcal{T}_h)$, then recover a bubble correction $p_h^b\in M^b(\mathcal{T}_h)$ so that $p_h=p_h^c+p_h^b$. The authors provide a rigorous well-posedness and convergence analysis, demonstrate local mass conservation, and show that the velocity field produced by EPG leads to stable, bound-preserving transport; numerical experiments across multiple domains confirm optimal convergence and superior mass-conservation properties compared to standard CG. The approach yields efficiency advantages over DG and simpler analysis than EG, with strong practical relevance for coupled flow-transport simulations in porous media and potential extensions to fractured media.

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.

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

TL;DR

The paper tackles the lack of local mass conservation in standard continuous Galerkin methods for Darcy flow in porous media by introducing an enriched Petrov-Galerkin (EPG) method. The EPG framework enriches the CG trial space with bubble functions and the test space with piecewise constants, enabling a penalty-free formulation and a decoupled two-step pressure solve: first compute in , then recover a bubble correction so that . The authors provide a rigorous well-posedness and convergence analysis, demonstrate local mass conservation, and show that the velocity field produced by EPG leads to stable, bound-preserving transport; numerical experiments across multiple domains confirm optimal convergence and superior mass-conservation properties compared to standard CG. The approach yields efficiency advantages over DG and simpler analysis than EG, with strong practical relevance for coupled flow-transport simulations in porous media and potential extensions to fractured media.

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.
Paper Structure (11 sections, 9 theorems, 82 equations, 16 figures, 1 algorithm)

This paper contains 11 sections, 9 theorems, 82 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical model and the EPG method
  3. Governing equations
  4. The EPG method
  5. Analysis of the EPG method
  6. Continuity of the bilinear forms
  7. Coercivity of the bilinear forms
  8. Convergence analysis
  9. Local mass conservation and application to the species transport system
  10. Numerical experiments
  11. Conclusion

Key Result

Theorem 3.1

The bilinear form (bilinear) is continuous, i.e., where $C>0$ is a constant independent of the mesh size $h$.

Figures (16)

  • Figure 5.1: (Example \ref{['example1']}) The solutions of pressure and Darcy velocity. Top-left: P1-CG. Top-middle: P2-CG. Top-right: P3-CG. Bottom-left: P1-EPG. Bottom-middle: P2-EPG. Bottom-right: P3-EPG.
  • Figure 5.2: (Example \ref{['example1']}) Relative errors. Top-bottom: P1, P2 and P3. Left: Energy error of pressure. Middle: $L^2$ error of velocity. Right: $L^2$ error of the normal component of velocity on the trace.
  • Figure 5.3: (Example \ref{['example1']}) Local mass conservation residual. Top-left: P1-CG. Top-middle: P2-CG. Top-right: P3-CG. Bottom-left: P1-EPG. Bottom-middle: P2-EPG. Bottom-right: P3-EPG.
  • Figure 5.4: (Example \ref{['example1']}) Simulations of concentration at time 4.5 based on the velocity from the CG method. Left: P1-CG. Middle: P2-CG. Right: P3-CG.
  • Figure 5.5: (Example \ref{['example1']}) Simulations of concentration based on the velocity from P3-EPG. Left-right: Simulations at time 0.2, 0.4 and 0.8.
  • ...and 11 more figures

Theorems & Definitions (21)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 11 more