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

Jongho Park, S. Majid Hassanizadeh

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.

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

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.

Paper Structure

This paper contains 18 sections, 8 theorems, 60 equations, 3 figures, 9 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. The Darcy--Forchheimer model
  3. The continuous problem
  4. Finite element approximations
  5. Reduction to a nearly semicoercive convex optimization problem
  6. Multilevel methods
  7. Multilevel subspace correction
  8. Convergence analysis
  9. Backtracking line search
  10. Full approximation scheme (FAS)
  11. Numerical results
  12. Augmented Lagrangian method
  13. Backtracking line search and FAS
  14. Dependence on epsilon and h, and coarsening factor
  15. Comparison with multilevel Newton--Krylov--Schwarz
  16. ...and 3 more sections

Key Result

Proposition 2.1

The function $\boldsymbol{u}_h \in X_h$ is a primal solution of mixed_FEM if and only if it solves the following convex optimization problem with a linear constraint: where the functional $F$ is defined in F, and $g_h$ is the $L^2 (\Omega)$-orthogonal projection of $g$ onto $M_h$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Multilevel mesh hierarchy $\{ \mathcal{T}_j \}_{j=1}^J$ for a rectangular grid $\mathcal{T}_h$ ($J = 4$). For each vertex $x_{j,k}$ of the mesh $\mathcal{T}_j$ (depicted in red), the blue region represents the corresponding subdomain $\Omega_{j,k}$.
  • Figure 1: Convergence curves of the relative energy error $\frac{F^{\epsilon}(\boldsymbol{u}^{(n)}) - \min F^{\epsilon}}{|\min F^{\epsilon}|}$ for the multilevel method (\ref{['Alg:multilevel']}) with step sizes $\tau = 2^{-2}, 2^{-3}, 2^{-4}$, the backtracking variant (\ref{['Alg:multilevel_backt']}), and the FAS variant (\ref{['Alg:multilevel_FAS']}), for various values of the augmented Lagrangian parameter $\epsilon$. Solid, dashed, and dash--dot lines correspond to Algorithms \ref{['Alg:multilevel']}, \ref{['Alg:multilevel_backt']}, and \ref{['Alg:multilevel_FAS']}, respectively (\ref{['Ex:Ex1']}, $\beta = 30$, $h = 2^{-8}$).
  • Figure 2: Convergence curves of the relative energy error $\frac{F^{\epsilon}(\boldsymbol{u}^{(n)}) - \min F^{\epsilon}}{|\min F^{\epsilon}|}$ for the multilevel method (\ref{['Alg:multilevel']}) with step sizes $\tau = 2^{-2}, 2^{-3}, 2^{-4}$, the backtracking variant (\ref{['Alg:multilevel_backt']}), and the FAS variant (\ref{['Alg:multilevel_FAS']}), for various values of the augmented Lagrangian parameter $\epsilon$. Solid, dashed, and dash--dot lines correspond to Algorithms \ref{['Alg:multilevel']}, \ref{['Alg:multilevel_backt']}, and \ref{['Alg:multilevel_FAS']}, respectively (\ref{['Ex:Ex2']}, $\beta = 30$, $h = 2^{-8}$).

Theorems & Definitions (21)

  • Proposition 2.1
  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Remark 4.5
  • Remark 4.6
  • ...and 11 more