Numerical discretization of a Darcy-Forchheimer problem coupled with a singular heat equation

TL;DR

This paper develops and analyzes a finite element discretization for a coupled Darcy-Forchheimer and singular heat equation problem, including convergence, error estimation, and adaptive methods with numerical validation.

Contribution

It introduces a finite element scheme for a complex coupled PDE system with singular heat source and provides both a priori and a posteriori error analysis.

Findings

Proved existence of solutions for the coupled problem.

Developed an adaptive finite element method with error control.

Validated the approach through numerical experiments.

Abstract

In Lipschitz domains, we study a Darcy-Forchheimer problem coupled with a singular heat equation by a nonlinear forcing term depending on the temperature. By singular we mean that the heat source corresponds to a Dirac measure. We establish the existence of solutions for a model that allows a diffusion coefficient in the heat equation depending on the temperature. For such a model, we also propose a finite element discretization scheme and provide an a priori convergence analysis. In the case that the aforementioned diffusion coefficient is constant, we devise an a posteriori error estimator and investigate reliability and efficiency properties. We conclude by devising an adaptive loop based on the proposed error estimator and presenting numerical experiments.