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Nonconforming virtual element method for an incompressible miscible displacement problem in porous media

Sarvesh Kumar, Devika Shylaja

TL;DR

The paper addresses a priori error estimates for miscible displacement in porous media modeled by a coupled nonlinear elliptic-parabolic system. It develops a mixed virtual element framework with $H(\mathrm{div})$-conforming VEM for velocity, a non-conforming VEM for concentration, and pressure discretization by discontinuous polynomials, all coupled with backward Euler time stepping. It proves optimal error estimates for the concentration as well as velocity and pressure, employing a projection-based analysis and Aubin–Nitsche duality to bound the $L^2$-concentration error, and provides a fully discrete, decoupled scheme with stability guarantees. Numerical experiments on general polygonal meshes with $k=0$ corroborate the theoretical convergence rates and demonstrate robustness across mesh families and realistic test cases, highlighting the practical applicability of nonconforming VEM for miscible flows in porous media.

Abstract

This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the $H(\rm{div})$ conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented.

Nonconforming virtual element method for an incompressible miscible displacement problem in porous media

TL;DR

The paper addresses a priori error estimates for miscible displacement in porous media modeled by a coupled nonlinear elliptic-parabolic system. It develops a mixed virtual element framework with -conforming VEM for velocity, a non-conforming VEM for concentration, and pressure discretization by discontinuous polynomials, all coupled with backward Euler time stepping. It proves optimal error estimates for the concentration as well as velocity and pressure, employing a projection-based analysis and Aubin–Nitsche duality to bound the -concentration error, and provides a fully discrete, decoupled scheme with stability guarantees. Numerical experiments on general polygonal meshes with corroborate the theoretical convergence rates and demonstrate robustness across mesh families and realistic test cases, highlighting the practical applicability of nonconforming VEM for miscible flows in porous media.

Abstract

This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented.
Paper Structure (14 sections, 12 theorems, 122 equations, 9 figures, 10 tables)

This paper contains 14 sections, 12 theorems, 122 equations, 9 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Weak formulation
  3. Main result
  4. The virtual element method
  5. Discrete spaces
  6. Semi-discrete formulation
  7. Fully discrete formulation
  8. Error estimates
  9. Error analysis for concentration
  10. Numerical results
  11. Example 1
  12. Example 2
  13. Example 3
  14. Appendix

Key Result

Lemma 2.1

The following properties hold for the bilinear forms in defn.bilinear-defn.bilinear_1Veiga_miscibledisplacement_2021: where $\eta_\mathcal{D}$ is a positive constant depending only on $d_m$, $d_\ell$, and $d_t$.

Figures (9)

  • Figure 1: Triangular Mesh
  • Figure 2: Square Mesh
  • Figure 3: Concave Mesh
  • Figure 4: Structured Voronoi Mesh
  • Figure 5: Random Voronoi Mesh
  • ...and 4 more figures

Theorems & Definitions (15)

  • Lemma 2.1: Properties of the bilinear forms
  • Lemma 3.1: Approximation properties
  • proof
  • Lemma 3.2: Properties of the discrete bilinear forms
  • Lemma 4.1: Auxiliary result
  • Theorem 4.2: Error for velocity and pressure
  • Remark 4.3
  • Lemma 4.4
  • Lemma 4.5
  • proof
  • ...and 5 more