*Proceeding Paper* **On the Adaptive Numerical Solution to the Darcy–Forchheimer Model †**

**María González \* and Hiram Varela \***

Departamento de Matemáticas and CITIC, Universidade da Coruña, Campus de Elviña s/n, 15071 A Coruña, Spain

**\*** Correspondence: maria.gonzalez.taboada@udc.es (M.G.); hiram.varela@udc.es (H.V.)

† Presented at the 4th XoveTIC Conference, A Coruña, Spain, 7–8 October 2021.

**Abstract:** We considered a primal-mixed method for the Darcy–Forchheimer boundary value problem. This model arises in fluid mechanics through porous media at high velocities. We developed an a posteriori error analysis of residual type and derived a simple a posteriori error indicator. We proved that this indicator is reliable and locally efficient. We show a numerical experiment that confirms the theoretical results.

**Keywords:** Darcy–Forchheimer; mixed finite element; a posteriori error estimates

### **1. Introduction**

The Darcy–Forchheimer model constitutes an improvement of the Darcy model which can be used when the velocity is high [1]. It is useful for simulating several physical phenomena, remarkably including fluid motion through porous media, as in petroleum reservoirs, water aquifers, blood in tissues or graphene nanoparticles through permeable materials. Let Ω be a bounded, simply connected domain in R<sup>2</sup> with a Lipschitz-continuous boundary *∂*Ω. The problem reads as follows: given known functions **g** and *f* , find the velocity **u** and the pressure *p* such that

$$\begin{cases} -\frac{\mu}{\rho} \mathbf{K}^{-1} \mathbf{u} + \frac{\beta}{\rho} |\mathbf{u}| \mathbf{u} + \nabla p &=& \mathbf{g} \quad \text{in } \Omega, \\\\ \nabla \cdot \mathbf{u} &=& f \quad \text{in } \Omega, \end{cases} \tag{1}$$

$$\mathbf{u} \cdot \mathbf{n} = 0 \quad \text{on } \partial \Omega,$$

where *μ* is the dynamic viscosity, *ρ* denotes the fluid density, *β* is the *Forchheimer number K* denotes the permeability tensor, **g** represents gravity, *f* is compressibility, and **n** is the unit outward normal vector to *∂*Ω.

We make use of the finite element method to approximate the solution of problem (1). We present the approach by Girault and Wheeler [1], who introduced the primal formulation, in which the term ∇ · **u** undergoes weakening by integration by parts. It is shown in [1] that problem (1) has a unique solution in the space *X* × *M*, where *X* := [*L*3(Ω)]<sup>2</sup> and *<sup>M</sup>* :<sup>=</sup> *<sup>W</sup>*1,3/2(Ω) <sup>∩</sup> *<sup>L</sup>*<sup>2</sup> <sup>0</sup>(Ω) (we use the standard notations for Lebesgue and Sobolev spaces).
