Spectral discretization of Darcy's equations coupled with the heat equation

Abstract

In this paper we consider the heat equation coupled with Darcy's law with a nonlinear source term describing heat production due to an exothermic chemical reaction. Existence and uniqueness of a solution are established. Next, a spectral discretization of the problem is presented and thoroughly analysed. Finally, we present some numerical experiments which confirm the interest of the discretization.Résumé: Dans cet article, nous considérons l'équation de la chaleur couplée avec la loi de Darcy avec un terme source non linéaire décrivant la production de chaleur dueà une réaction chimique exothermique. L'existence et l'unicité d'une solution sontétablies. Ensuite, une discrétisation par méthode spectrale du problème est décrite et analysée en détail. Enfin, nous présentons quelques expériences numériques qui confirment l'intérêt de la discrétisation.

Finite element methods for the Darcy-Forchheimer problem coupled with the convection-diffusion-reaction problem

“…For the coupled problem of Darcy's law with the heat equation, we can refer to [6] where the coupled problem is treated using the spectral method. The authors in [4] and [12] considered the same stationary system but coupled with a nonlinear viscosity that depends on the temperature.…”

“…In this case, the Darcy-Forchheimer system is coupled with the concentration C of fluid with the external force f . We mention that the work of [6] use the spectral method to treat a coupled system very close to (P ) where the Darcy-Forchheimer equation is replaced by the Darcy one. It turns out that the non-linear term appearing in the first equation of (P ) makes the treatment of the coupled system more complex.…”

Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

“…The literature of heat convection in a liquid medium whose motion is described by the Navier–Stokes or Darcy equations coupled with the heat equation under Dirichlet boundary condition is rich and we refer the reader among others to .…”

Iterative methods for Stokes flow under nonlinear slip boundary condition coupled with the heat equation

“…The study of heat convection in a liquid medium whose motion is described by the Navier Stokes or Darcy equations coupled with the heat equation under Dirichlet boundary condition have been investigated in many publications (see [8,9,10,11] just to mentioned a few). In [8,9,11], the continuous and approximate variational formulations for the Navier-Stokes/Darcy equations coupled with the heat equation are formulated and thorough discussion about the existence and uniqueness for both formulations are provided. The convergence of the approximate solution is also made clear.…”