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A posteriori error analysis of a mixed FEM for the coupled Brinkman-Forchheimer/Darcy problem

Sergio Caucao, Paulo Zúñiga

TL;DR

This work addresses robust a posteriori error control for the 2D mixed finite element discretization of the coupled Brinkman--Forchheimer and Darcy problem with transmission across an interface. It develops a residual-based estimator, expressed via local indicators on the Brinkman and Darcy regions and the interface, and proves both reliability and efficiency by exploiting inf-sup stability, strong monotonicity, Helmholtz decompositions, and interpolation estimates. The main contributions are the first reliable and efficient a posteriori error analysis for this coupled system in 2D, together with numerical demonstrations showing effective adaptive mesh refinement and recovery of optimal convergence rates in heterogeneous porous media. The practical impact lies in enabling accurate, efficient simulations of complex flows in composite porous media through targeted mesh refinement guided by the estimator. Key results include the global estimator $\\Theta_{\\text{BFD}}$ and its local components, plus rigorous bounds linking the estimator to the true discretization error.

Abstract

We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of the associated conforming mixed finite element scheme. For the reliability analysis, due to the nonlinear nature of the problem, we make use of the inf-sup condition and the strong monotonicity of the operators involved, along with a stable Helmholtz decomposition in Hilbert spaces and local approximation properties of the Raviart--Thomas and Clément interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of flow through a heterogeneous porous medium is considered.

A posteriori error analysis of a mixed FEM for the coupled Brinkman-Forchheimer/Darcy problem

TL;DR

This work addresses robust a posteriori error control for the 2D mixed finite element discretization of the coupled Brinkman--Forchheimer and Darcy problem with transmission across an interface. It develops a residual-based estimator, expressed via local indicators on the Brinkman and Darcy regions and the interface, and proves both reliability and efficiency by exploiting inf-sup stability, strong monotonicity, Helmholtz decompositions, and interpolation estimates. The main contributions are the first reliable and efficient a posteriori error analysis for this coupled system in 2D, together with numerical demonstrations showing effective adaptive mesh refinement and recovery of optimal convergence rates in heterogeneous porous media. The practical impact lies in enabling accurate, efficient simulations of complex flows in composite porous media through targeted mesh refinement guided by the estimator. Key results include the global estimator and its local components, plus rigorous bounds linking the estimator to the true discretization error.

Abstract

We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of the associated conforming mixed finite element scheme. For the reliability analysis, due to the nonlinear nature of the problem, we make use of the inf-sup condition and the strong monotonicity of the operators involved, along with a stable Helmholtz decomposition in Hilbert spaces and local approximation properties of the Raviart--Thomas and Clément interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of flow through a heterogeneous porous medium is considered.

Paper Structure

This paper contains 12 sections, 18 theorems, 124 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary notations
  3. The model problem and its variational formulation
  4. The coupled Brinkman--Forchheimer/Darcy problem
  5. The variational formulation
  6. The conforming finite element method
  7. A residual-based a posteriori error estimator
  8. Reliability of the a posteriori error estimator
  9. Efficiency of the a posteriori error estimator
  10. Numerical results
  11. Preliminaries for reliability
  12. Preliminaries for efficiency

Key Result

Theorem 3.1

There exists a constant $C_{\tt rel}>0$, independent of $h$, such that

Figures (6)

  • Figure 2.1: Sketch of a 2D geometry of the coupled Brinkman--Forchheimer/Darcy model
  • Figure 4.1: [ Example 2] Log-log plot of $\mathsf{e}(\vec{{\boldsymbol\sigma}})$ vs. ${\mathtt{DoF}}$ for quasi-uniform/adaptive refinements.
  • Figure 4.2: [ Example 2] Initial mesh, second velocity component and pressure field in the whole domain.
  • Figure 4.3: [ Example 2] Three snapshots of adapted meshes according to the indicator $\Theta_{\mathtt{BFD}}$.
  • Figure 4.4: [ Example 3] Initial mesh, second velocity component and magnitude of the velocity in the whole domain.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Lemma 3.8
  • Lemma 3.9
  • Lemma 3.10
  • ...and 8 more