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Finite element approximation for a convective Brinkman--Forchheimer problem coupled with a heat equation

Gilberto Campaña, Pablo Muñoz, Enrique Otarola

Abstract

We investigate a convective Brinkman--Forchheimer problem coupled with a heat equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We also propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.

Finite element approximation for a convective Brinkman--Forchheimer problem coupled with a heat equation

Abstract

We investigate a convective Brinkman--Forchheimer problem coupled with a heat equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We also propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.
Paper Structure (22 sections, 16 theorems, 104 equations, 3 figures, 1 algorithm)

This paper contains 22 sections, 16 theorems, 104 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminary remarks
  3. Notation
  4. A convective Brinkman--Forchheimer problem
  5. Existence
  6. Uniqueness
  7. A nonlinear heat equation
  8. The coupled problem
  9. Main assumptions
  10. Weak solution
  11. Existence of solutions
  12. Uniqueness of solutions
  13. Finite element approximation
  14. Notation and basic components
  15. Finite element spaces
  16. ...and 7 more sections

Key Result

Theorem 2

\newlabelth:BDF_exist0 There exists at least one solution $(\mathbf{u},\mathsf{p})\in \mathbf{H}_0^1(\Omega)\times L_0^2(\Omega)$ for problem eq:problem_variational_forms. Moreover, $(\mathbf{u},\mathsf{p})$ satisfies the bound where $\beta$ corresponds to the constant in eq:infsup_cont.

Figures (3)

  • Figure 1: Example 1. Performance of the developed finite element discretization scheme: Taylor--Hood approximation for the velocity and pressure variables and the space of continuous piecewise quadratic functions to approximate the temperature variable. We present experimental convergence rates for the errors that occur when approximating the velocity field, pressure, and temperature variables in appropriate norms for $s=3.0$ (A.1), $s=3.5$ (A.2), and $s=4.0$ (A.3).
  • Figure 2: Boundary conditions for the cavity flow problem in the domain $\Omega=(0,1)^2$.
  • Figure 3: Example 2. Streamlines of the velocity field obtained for $s=3.0$ (C.1), $s=3.5$ (C.2), and $s=4.0$ (C.3), the contour lines of the pressure for $s=3.0$ (C.4), $s=3.5$ (C.5), and $s=4.0$ (C.6), and temperature variable for $s=3.0$ (C.7), $s=3.5$ (C.8), $s=4.0$ (C.9).

Theorems & Definitions (32)

  • Remark 1: boundedness of $\mathrm{I}_s$
  • Theorem 2: existence and stability bound
  • Proof 1
  • Theorem 3: uniqueness for small data
  • Proof 2
  • Theorem 4: existence
  • Proof 3
  • Theorem 5: uniqueness for small data
  • Proof 4
  • Remark 6: $d=2$
  • ...and 22 more