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Robust stability and preconditioning of Darcy-Forchheimer equations

Rishi Das, Harsha Hutridurga, Amiya K. Pani, Ricardo Ruiz-Baier

TL;DR

This work develops a parameter-robust analysis for the nonlinear Darcy--Forchheimer equations in porous media and introduces an operator-preconditioning framework that yields robust block preconditioners for the linearized system. Using parameter-weighted norms, Minty--Browder theory, and carefully constructed lifting operators, the authors establish continuous and discrete well-posedness with convergence rates that are independent of the permeability $\kappa$ and Forchheimer coefficient $\mathtt{F}$. They formulate a mixed finite element discretization with Raviart--Thomas velocity and a nonconforming pressure, proving quasi-optimal error estimates and discrete inf-sup stability that are robust to model parameters. The paper also proposes a variable operator preconditioner for the Newton linearization, with both velocity and pressure blocks designed to maintain uniform spectral bounds as $\kappa$ and $\mathtt{F}$ vary, and substantiates the approach with numerical experiments showing stable performance across extreme parameter regimes. Overall, the results offer a scalable, robust framework for nonlinear saddle-point problems in porous media flow and suitable coupling to multiphysics models.

Abstract

We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design efficient block preconditioners for the linearised system, that exhibit robustness with respect to the coefficients that modulate permeability and inertia of the system. The properties of the formulation (parameter and mesh-size independence of the convergence rates) are illustrated by means of several numerical examples.

Robust stability and preconditioning of Darcy-Forchheimer equations

TL;DR

This work develops a parameter-robust analysis for the nonlinear Darcy--Forchheimer equations in porous media and introduces an operator-preconditioning framework that yields robust block preconditioners for the linearized system. Using parameter-weighted norms, Minty--Browder theory, and carefully constructed lifting operators, the authors establish continuous and discrete well-posedness with convergence rates that are independent of the permeability and Forchheimer coefficient . They formulate a mixed finite element discretization with Raviart--Thomas velocity and a nonconforming pressure, proving quasi-optimal error estimates and discrete inf-sup stability that are robust to model parameters. The paper also proposes a variable operator preconditioner for the Newton linearization, with both velocity and pressure blocks designed to maintain uniform spectral bounds as and vary, and substantiates the approach with numerical experiments showing stable performance across extreme parameter regimes. Overall, the results offer a scalable, robust framework for nonlinear saddle-point problems in porous media flow and suitable coupling to multiphysics models.

Abstract

We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design efficient block preconditioners for the linearised system, that exhibit robustness with respect to the coefficients that modulate permeability and inertia of the system. The properties of the formulation (parameter and mesh-size independence of the convergence rates) are illustrated by means of several numerical examples.

Paper Structure

This paper contains 16 sections, 11 theorems, 91 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Scope
  3. Outline
  4. Model problem
  5. Notation and preliminaries
  6. Strong form of the governing equations
  7. Continuous robust solvability
  8. Local Lipschitz continuity
  9. Uniformly strong monotonicity of $\{A+C\}(\bullet+\boldsymbol{w})$
  10. Inf-sup stability
  11. Inf-sup stability
  12. Mixed finite element method and robust solvability
  13. Robust discrete solvability
  14. A priori error analysis
  15. Preconditioning for the linearised Darcy--Forchheimer equations
  16. ...and 1 more sections

Key Result

Theorem 1

\newlabelth:abstract0 Let $X,Y$ be two reflexive Banach spaces, $\mathcal{A}:X\to X'$ a nonlinear map, $\mathcal{B}:X\to Y'$ a linear and bounded operator, and denote by $Z:= \{ v\in X: \mathcal{B}(v) = 0\}$ the kernel of $\mathcal{B}$. Assume that Then, for each $(f, g) \in X' \times Y'$ there exists a unique $(u, p) \in X \times Y$ such that Moreover, there exists $C > 0$, depending only on $

Figures (3)

  • Figure 1: Example 1. Streamlines of the approximate velocity (left) and pressure distribution (right) for the accuracy test of the Darcy--Forchheimer equations with unit parameters.
  • Figure 2: Example 2. Sample of filtration velocity magnitude, pressure profile, and heterogeneous permeability, for the convergence test with no known analytical solution.
  • Figure 3: Example 3. Streamlines of the approximate velocity coloured by velocity magnitude (left) and contours of pressure distribution (right), for the flow across obstacles using the Darcy--Forchheimer equations with anisotropic permeability. A second-order scheme ($k=1$) is used.

Theorems & Definitions (22)

  • Theorem 1: Abstract setting for well-posedness
  • Lemma 2
  • Proof 1
  • Theorem 3: Parameter-robust well-posedness
  • Proof 2
  • Remark 4.1
  • Lemma 1
  • Proof 3
  • Lemma 2: Discrete inf-sup condition
  • Proof 4
  • ...and 12 more