A Hybridizable Discontinuous Galerkin Method for the Miscible Displacement Problem Under Minimal Regularity

TL;DR

This paper introduces a hybridizable discontinuous Galerkin method combined with backward Euler time stepping for the miscible displacement problem, proving convergence under minimal regularity assumptions and demonstrating optimal numerical performance.

Contribution

It develops a novel numerical scheme that ensures convergence for low regularity solutions, expanding the applicability of DG methods to complex fluid displacement problems.

Findings

Proves convergence of the method under minimal regularity assumptions.

Demonstrates optimal convergence rates for smooth solutions.

Shows convergence for problems with low regularity.

Abstract

A numerical method based on the hybridizable discontinuous Galerkin method in space and backward Euler in time is formulated and analyzed for solving the miscible displacement problem. Under low regularity assumptions, convergence is established by proving that, up to a subsequence, the discrete pressure, velocity and concentration converge to a weak solution as the mesh size and time step tend to zero. The analysis is based on several key features: an H(div) reconstruction of the velocity, the skew-symmetrization of the concentration equation, the introduction of an auxiliary variable and the definition of a new numerical flux. Numerical examples demonstrate optimal rates of convergence for smooth solutions, and convergence for problems of low regularity.