A Posteriori error estimates for Darcy-Forchheimer's problem coupled with the convection-diffusion-reaction equation

TL;DR

This paper develops and analyzes a posteriori error estimates for a coupled Darcy-Forchheimer and convection-diffusion-reaction problem, providing reliable error indicators and validating them through numerical experiments.

Contribution

It introduces new a posteriori error estimates for a nonlinear coupled PDE system, including both linearization and discretization error indicators.

Findings

Optimal error bounds are established under regularity assumptions.

Two types of error indicators are proposed and validated.

Numerical results confirm the effectiveness of the error estimators.

Abstract

In this work we derive a posteriori error estimates for the convection-diffusion-reaction equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending on the concentration of the fluid. We introduce the variational formulation associated to the problem, and discretize it by using the finite element method. We prove optimal a posteriori errors with two types of calculable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we find upper and lower error bounds under additional regularity assumptions on the exact solutions. Finally, numerical computations are performed to show the effectiveness of the obtained error indicators.