Parallel multilevel methods for solving the Darcy--Forchheimer model based on a nearly semicoercive formulation

High-velocity fluid flow through porous media is modeled by prescribing a nonlinear relationship between the flow rate and the pressure gradient, called the Darcy–Forchheimer equation. This paper is concerned with the analysis of parallel multilevel methods for solving the Darcy–Forchheimer model. We begin by reformulating the Darcy–Forchheimer model as a nearly semicoercive convex optimization problem via the augmented Lagrangian method. Building on this formulation, we develop a parallel multilevel method, also known as a multilevel additive Schwarz method, within the framework of subspace correction for nearly semicoercive convex problems, yielding a theoretically supported and computationally efficient solver for the Darcy–Forchheimer model. The convergence analysis establishes robustness with respect to the augmented Lagrangian parameter $\epsilon$. To further enhance convergence, we incorporate a backtracking line search and a full approximation scheme. Numerical results support the theoretical findings and demonstrate the effectiveness of the proposed approach.