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. The proposed method exhibits robustness with respect to both the nearly semicoercive nature of the problem and the size of the discretized system. To further enhance convergence, we incorporate a backtracking line search scheme and a full approximation scheme. Numerical results validate the theoretical findings and demonstrate the effectiveness and superiority of the proposed approach.