Convergence analysis of a nonconforming virtual element method for compressible miscible displacement problems in porous media

TL;DR

This paper analyzes the convergence of a nonconforming virtual element method for simulating the flow of compressible fluids in porous media, providing error estimates and validating them with numerical results.

Contribution

It introduces a novel combination of conforming and nonconforming virtual element methods for different variables in compressible flow problems, with proven error bounds.

Findings

Error estimates for velocity, pressure, and concentration are established.

Numerical results confirm the theoretical error bounds.

The method effectively models compressible miscible displacement in porous media.

Abstract

This article presents a priori error estimates of the miscible displacement of one compressible fluid by another in a porous medium. 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. Error estimates are established for velocity, pressure and concentration. The article also includes numerical results that validate the theoretical estimates.