TITLE:
Optimal Penalization and Nonlinear Solver Convergence for a DG-Based Richards’ Equation Model of Variably Saturated Flows
AUTHORS:
Camille Poussel, Mehmet Ersoy, Yolhan Mannes, Aimed Ajroud, Frédéric Golay
KEYWORDS:
Porous Media, Richards Equation, Discontinuous Galerkin, Backward Differentiation Method, Incomplete Interior Penalty Galerkin (IIPG), Broken Sobolev Space, Picard’s Fixed Point, Minimal Regularity Solution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.11,
November
26,
2025
ABSTRACT: In this article, we study the convergence of an IIPG (Incomplete Interior Penalty Galerkin) Discontinuous Galerkin numerical method for the Richards equation. The Richards equation is a degenerate parabolic nonlinear equation for modeling flows in porous media with variable saturation. The numerical solution of this equation is known to be difficult to calculate numerically, due to the abrupt displacement of the wetting front, mainly as a result of highly nonlinear hydraulic properties. As time scales are slow, implicit numerical methods are required, and the convergence of nonlinear solvers is very sensitive. We propose an original method to ensure convergence of the numerical solution to the exact Richards solution, using a technique of auto-calibration of the penalty parameters derived from the Galerkin Discontinuous method. The method is constructed using nonlinear 1D and 2D general elliptic problems. We show that the numerical solution converges toward the unique solution of the continuous problem under certain conditions on the penalty parameters. Then, we numerically demonstrate the efficiency and robustness of the method through test cases with analytical solutions, laboratory test cases, and large-scale simulations.