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

TL;DR

This paper develops a parallel multilevel additive Schwarz method for solving the nonlinear Darcy--Forchheimer model, reformulated as a nearly semicoercive convex optimization problem, demonstrating robustness and efficiency through numerical validation.

Contribution

It introduces a novel parallel multilevel method tailored for nearly semicoercive convex problems derived from Darcy--Forchheimer equations, with enhanced convergence techniques.

Findings

Method is robust to problem semicoercivity and system size.

Incorporating line search and full approximation improves convergence.

Numerical results confirm theoretical robustness and efficiency.

Abstract

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.