Darcy's problem coupled with the heat equation under singular forcing: analysis and discretization

TL;DR

This paper investigates the existence and numerical approximation of solutions for a coupled Darcy and heat equation model with singular forcing, including analysis of convergence and error estimation.

Contribution

It introduces a finite element method for the coupled problem with singular forcing and provides convergence analysis and an a posteriori error estimator.

Findings

Finite element solution converges under certain conditions.

A reliable and efficient a posteriori error estimator is developed.

Numerical examples validate the theoretical results.

Abstract

We study the existence of solutions for Darcy's problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The studied model allows thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case that the thermal diffusion is constant, we propose an a posteriori error estimator and investigate reliability and efficiency properties. We illustrate the theory with numerical examples.