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Robust finite element methods and solvers for the Biot--Brinkman equations in vorticity form

Ruben Caraballo, Chansophea Wathanak In, Alberto F. Martín, Ricardo Ruiz-Baier

Abstract

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.

Robust finite element methods and solvers for the Biot--Brinkman equations in vorticity form

Abstract

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
Paper Structure (15 sections, 9 theorems, 79 equations, 4 figures, 3 tables)

This paper contains 15 sections, 9 theorems, 79 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem and its weak formulation
  3. Preliminaries
  4. The governing equations
  5. Weak formulation
  6. Solvability analysis
  7. Preliminaries
  8. Parameter-robust well-posedness
  9. Operator preconditioning
  10. Analysis of a finite element method
  11. Numerical verification
  12. Accuracy tests
  13. Robustness of the convergence rates
  14. Preconditioner robustness with respect to model parameters
  15. Concluding remarks

Key Result

theorem 3.1

Let $(E_1, \|\cdot\|_{E_1})$ be a reflexive Banach space, $(E_2,\|\cdot\|_{E_2})$ a Banach space, and $T:E_1\rightarrow E_2'$ a bounded, linear form satisfying the followings conditions:

Figures (4)

  • Figure 5.1: Accuracy test in 3D. Approximate solutions of the Biot--Brinkman equations on a relatively coarse mesh. Displacements on the deformed configuration, velocity streamlines, vorticity vectors, total and fluid pressures computed with the lowest-order method.
  • Figure 5.2: Error convergence curves in 3D for $k=0$ and different combinations of the physical parameters values. The errors are measured in the weighted norm that leads to the definition of $\mathcal{B}_3$.
  • Figure 5.3: Comparison of parameter robustness for preconditioners $\mathcal{B}_1$, $\mathcal{B}_2$, and $\mathcal{B}_3$.
  • Figure 5.4: Comparison of parameter robustness for preconditioners $\mathcal{B}_1$, $\mathcal{B}_2$, and $\mathcal{B}_3$.

Theorems & Definitions (18)

  • theorem 3.1
  • remark 3.1
  • lemma 3.1
  • proof
  • lemma 3.2
  • lemma 3.3
  • proof
  • theorem 3.2
  • proof
  • lemma 4.1
  • ...and 8 more