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Conforming and discontinuous discretizations of non-isothermal Darcy-Forchheimer flows

Stefano Bonetti, Michele Botti, Paola F. Antonietti

TL;DR

The paper tackles the non-isothermal Darcy–Forchheimer–heat system by developing two discretizations: a fully discontinuous Galerkin (dG-dG-dG) scheme and a Raviart–Thomas based RT-dG-dG scheme for velocity, both coupled to a discontinuous pressure and temperature field. A fixed-point linearization is employed to manage nonlinearities, accompanied by a unified stability analysis, existence and uniqueness results, and convergence of the iterative solver. The authors validate the methods through comprehensive 2D and 3D numerical experiments, including convergence tests and advection-dominated heat transport scenarios, demonstrating robust error decay and practical performance on polytopal meshes. The work provides a solid theoretical and computational framework for efficient, reliable simulation of high-velocity, nonlinear DF flows coupled to heat transport with potential geophysical applications.

Abstract

We present and analyze in a unified setting two schemes for the numerical discretization of a Darcy-Forchheimer fluid flow model coupled with an advection-diffusion equation modeling the temperature distribution in the fluid. The first approach is based on fully discontinuous Galerkin discretization spaces. In contrast, in the second approach, the velocity is approximated in the Raviart-Thomas space, and the pressure and temperature are still piecewise discontinuous. A fixed-point linearization strategy, naturally inducing an iterative splitting solution, is proposed for treating the nonlinearities of the problem. We present a unified stability analysis and prove the convergence of the iterative algorithm under mild requirements on the problem data. A wide set of two- and three-dimensional simulations is presented to assess the error decay and demonstrate the practical performance of the proposed approaches in physically sound test cases.

Conforming and discontinuous discretizations of non-isothermal Darcy-Forchheimer flows

TL;DR

The paper tackles the non-isothermal Darcy–Forchheimer–heat system by developing two discretizations: a fully discontinuous Galerkin (dG-dG-dG) scheme and a Raviart–Thomas based RT-dG-dG scheme for velocity, both coupled to a discontinuous pressure and temperature field. A fixed-point linearization is employed to manage nonlinearities, accompanied by a unified stability analysis, existence and uniqueness results, and convergence of the iterative solver. The authors validate the methods through comprehensive 2D and 3D numerical experiments, including convergence tests and advection-dominated heat transport scenarios, demonstrating robust error decay and practical performance on polytopal meshes. The work provides a solid theoretical and computational framework for efficient, reliable simulation of high-velocity, nonlinear DF flows coupled to heat transport with potential geophysical applications.

Abstract

We present and analyze in a unified setting two schemes for the numerical discretization of a Darcy-Forchheimer fluid flow model coupled with an advection-diffusion equation modeling the temperature distribution in the fluid. The first approach is based on fully discontinuous Galerkin discretization spaces. In contrast, in the second approach, the velocity is approximated in the Raviart-Thomas space, and the pressure and temperature are still piecewise discontinuous. A fixed-point linearization strategy, naturally inducing an iterative splitting solution, is proposed for treating the nonlinearities of the problem. We present a unified stability analysis and prove the convergence of the iterative algorithm under mild requirements on the problem data. A wide set of two- and three-dimensional simulations is presented to assess the error decay and demonstrate the practical performance of the proposed approaches in physically sound test cases.

Paper Structure

This paper contains 19 sections, 9 theorems, 82 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Weak formulation
  4. Discretization
  5. Preliminaries
  6. The dG-dG-dG discrete formulation
  7. The RT-dG-dG discrete problem
  8. Linearization
  9. Theoretical analysis
  10. Stability estimates
  11. Existence
  12. Uniqueness
  13. Convergence of the fixed-point algorithm
  14. Numerical results
  15. Convergence test case in two-dimensions
  16. ...and 4 more sections

Key Result

Lemma 3.1

For all $\mathbf{v} \in \mathbf{V}_h^{\ell}$, for all $S \in V_h^{\ell}$ it holds:

Figures (8)

  • Figure 1: Test case of Section \ref{['sec:conv_test2D']}. Computed errors in $L^2$-norm (left) and energy-norms (right) versus $1/h$ (log-log scale).
  • Figure 2: Test case of Section \ref{['sec:conv_test2D']}. Computed errors in $L^2$-norm (left) and energy-norms (right) versus the polynomial approximation degree $\ell$ (semilog scale).
  • Figure 3: Test case of Section \ref{['sec:conv_test3D']}. Computed errors in $L^2$-norm (left) and energy-norms (right) versus $1/h$ (log-log scale).
  • Figure 4: Test case of Section \ref{['sec:advectionstep2D']}: computed velocity field (left), temperature field (center), and pressure field (right).
  • Figure 5: Test case of Section \ref{['sec:advectionstep3D']}: computed velocity field. The domain is clipped at $x=0.5$ and the four slices are at $y=0.95$ (top-left), $y=1.05$ (bottom-left), $z=0.95$ (top-right), and $z=1.05$ (bottom-right)
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 3.1: Polytopal regular mesh Cangiani2014CangianiDong2017
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.1
  • proof
  • ...and 12 more