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

TL;DR

This paper introduces and analyzes two numerical schemes for simulating non-isothermal Darcy-Forchheimer flows, focusing on stability, convergence, and practical performance through comprehensive simulations.

Contribution

It presents a unified analysis of two discretization methods, including a fixed-point linearization strategy, for non-isothermal Darcy-Forchheimer flow models.

Findings

Both schemes are stable and convergent under mild conditions.

Numerical experiments demonstrate effective error decay.

The methods perform well in realistic 2D and 3D test cases.

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.