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 $ε$. 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.