On the Adaptive Numerical Solution to the Darcy–Forchheimer Model †

: 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.


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 2 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 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)] 2 and M := W 1,3/2 (Ω) ∩ L 2 0 (Ω) (we use the standard notations for Lebesgue and Sobolev spaces).

Discrete Problem
To pose a discrete problem, we can use a family {T h } h>0 of conforming triangulations to divide the domainΩ such thatΩ = T∈T h T, ∀h, where h > 0 represents the mesh size. Here we follow [2] and choose the following conforming discrete subspaces of X and M, respectively: Then, the discrete problem consists in finding It is shown in [2] that problem (2) has a unique solution and that the sequence {(u h , p h )} h converges to the exact solution of problem (1) in X × M. Furthermore, under additional regularity assumptions on the exact solution, some error estimates were derived in [2].

Novel Error Estimator and Adaptive Algorithm
We denote by E Ω , E ∂Ω and E T , respectively, the sets of edges e belonging to the interior domain, the boundary and the element T; h e denotes the length of a particular edge e; and h T is the diameter of a given element T. We denote by J e (v) the jump of v across the edge e in the direction of n e , a fixed normal vector to side e. Finally, we use the operator On every triangle T ∈ T h , we propose the following a posteriori error indicator: We propose an adaptive algorithm based on the a posteriori error indicator θ. Given an initial mesh, we follow the iterative procedure described in Figure 1. Each new mesh is generated as suggested in [3].

Numerical Experiment
We performed several simulations in FreeFem++ [4], validating the theoretical results. Here we select an example on an L-shaped domain, Ω = (−1, 1) 2 \[0, 1] 2 , and focus on the data f and g so that the exact solution is Thus the solution has a singularity in pressure close to the line x = 1. Figure 2 shows the mesh refinement by the adaptive algorithm. Figure 3, bottom, represents the evolution with respect to degrees of freedom (DOF) of error and indicator; on the right, we can observe the evolution of the efficiency index with DOF.

Discussion
The adaptive algorithm was tested on an example with a singularity. From Figure 2 we can observe that the algorithm refined the mesh near the singularity, as expected. Since it is an academic example with a known solution, we could compute the exact error. The graphs in Figure 3 confirm that the error was lower for the adaptive refinement. Additionally, since the exact error and estimator followed close to parallel lines, we confirm that the indicator gives a consistent measure of the error. This could also be checked by the efficiency index, which is the ratio of indicator to exact total error.

Conflicts of Interest:
The authors declare no conflict of interest.