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

TL;DR

The paper addresses the miscible displacement problem in porous media with minimal regularity, where fully coupled flow and nonlinear transport pose significant analytical challenges. It develops a unconditionally stable Hybridizable Discontinuous Galerkin (HDG) method in space and backward Euler time stepping, featuring an H(div) velocity projection, a skew-symmetrized transport formulation, and a three-field reformulation with auxiliary variables to manage nonlinearity. The authors prove convergence (up to subsequences) to a weak solution as $h,\tau \to 0$, and establish a novel compactness framework including a discrete Aubin–Lions lemma for time-dependent HDG approximations. Numerical experiments confirm optimal convergence rates for smooth solutions and demonstrate robustness in low-regularity scenarios, with the H(div) velocity reconstruction shown to be essential for stability and accuracy. These results provide a rigorous foundation for HDG discretizations of coupled flow–transport problems under minimal regularity and guide practical implementation for subsurface transport simulations.

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.

Figures (6)

• Figure 1: $L^2$ norm of the error as a function of refinement level, for different polynomial degree $k$, for the velocity $\bm u_h$, the reconstructed velocity $\bm U_h$ and the pressure $p_h$ at the final time.
• Figure 2: $L^2$ norm of the error as a function of refinement level, for different polynomial degree $k$, for the concentration $c_h$ and the flux $\bm q_h$ at the final time.
• Figure 3: L-shape domain and source/sink functions set-up. Permeability is discontinuous and takes the values $\kappa_1$ in $\Omega_1$ and $\kappa_2$ in the rest of the domain.
• Figure 4: Surface plots of the approximate concentration at $t=3$ for $\tau = 0.1$ (left), $\tau = 0.05$ (middle), $\tau = 0.025$ (right). Top row: with $H(\text{div})$ reconstruction; bottom row: without $H(\text{div})$ reconstruction.
• Figure 5: Contour plots of the approximate concentration at $t=3$ for $\tau = 0.1$ (left), $\tau = 0.05$ (middle), $\tau = 0.025$ (right). Top row: with $H(\text{div})$ reconstruction; bottom row: without $H(\text{div})$ reconstruction. Overshoots (values above $1.001$) and undershoots (values below $-0.001$) are shaded in red and blue, respectively.

Theorems & Definitions (36)

• Definition 1: Weak formulation
• Remark 1
• Lemma 1: HDG Poincaré inequality
• Lemma 2
• Proof 1
• Theorem 3
• Proof 2
• Corollary 4
• Proof 3
• Theorem 5