Examples of 2D Darcy flows

The following two examples come from [BD].

1. Example : 2D Darcy flow, chessboard pressure

1.1. Input parameters

Notation	Quantity	Type	Unit
$\underline{\underline\kappa}$	Permeability	order 2 tensor	$m^2Pa^{-1}s^{-1}$
$f(x,y)$	Flux source term	scalar function	$s^{-1}$

Notation

Quantity

Type

Unit

$\underline{\underline\kappa}$

Permeability

order 2 tensor

$m^2Pa^{-1}s^{-1}$

$f(x,y)$

Flux source term

scalar function

$s^{-1}$

1.2. Model & Toolbox

We consider a 2D unit square $\Omega=[0,1]\times[0,1]$ whose boundary is denoted $\Gamma$ . The following problem is to be solved in $\Omega$ .

$\left\{\begin{array}{rll} \underline u+\underline{\underline\kappa}\nabla p=&0&\text{in }\Omega\\ \nabla\cdot\underline u=&f&\text{in }\Omega\\ \end{array}\right.$

with the additional condition $\int_\Omega f=0$ . The pressure is denoted $p$ and the velocity $\underline u$ . We assume the material permeability is constant, isotropic and unitary, that is $\underline{\underline\kappa}=\underline{\underline{I_d}}$ .

Let us define the source term $f=\nabla\cdot\underline u=-\Delta p$ to get the analytic solution $p(x,y)=sin(2\pi x)cos(2\pi y)$ . It yields $f(x,y)=8\pi^2sin(2\pi x)cos(2\pi y)$ .

This example runs within the Mixed Poisson toolbox with prescribed parameters.

1.3. Boundary conditions

We impose a Dirichlet boundary condition on the whole boundary : $p=sin(2\pi x)cos(2\pi y)\text{ on }\Gamma$ .

1.4. Configuration & geometry

GEO file

CFG file

JSON file

1.5. Convergence analysis

Table 1. P0 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	9.57939e-01	5.42923e-01	2.78594e-01	5.6416e-02	2.83271e-02
$\Vert u-u_h\Vert_{L^2}$	2.78314e-01	1.35505e-01	6.61506e-02	1.30739e-02	6.52889e-03

Table 2. P1 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	1.69091e-01	4.85275e-02	1.26349e-02	5.14523e-04	1.28986e-04
$\Vert u-u_h\Vert_{L^2}$	1.66947e-01	4.78222e-02	1.22767e-02	4.92702e-04	1.23431e-04

Table 3. P2 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	2.22396e-02	3.15292e-03	4.07591e-04	3.22962e-06	4.0602e-07
$\Vert u-u_h\Vert_{L^2}$	1.73431e-02	2.35603e-03	3.01594e-04	2.33871e-06	2.93291e-07

Table 4. P3 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	2.03629e-03	1.52963e-04	9.81156e-06	1.56186e-08	9.80369e-10
$\Vert u-u_h\Vert_{L^2}$	1.37478e-03	1.01811e-04	6.43878e-06	1.00732e-08	6.3051e-10

1.6. Output

The following output example is reproducible using feelpp_toolbox_mixed-poisson-model_2DP2 running on 12 cores with the previous .json and .cfg files on a mesh of typical size $h=0.05$ .

The screenshots are in order : the pressure field, the velocity magnitude field and the velocity field.

pressuremap

velocitymap

fluxfield

1.7. Example : Darcy flow, shower

The input parameters, model and toolbox are the same as in the previous example.

1.8. Boundary conditions

We impose a Dirichlet boundary condition on the whole boundary : $p=sin(y)sin(x)+xy^2-\frac16-\sin(1)(1-\cos(1))\text{ on }\Gamma$ .

1.9. Configuration & geometry

GEO file

CFG file

JSON file

1.10. Convergence analysis

Table 5. P0 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	5.34577e-02	2.79542e-02	1.42528e-02	2.85709e-03	1.43102e-03
$\Vert u-u_h\Vert_{L^2}$	4.78442e-02	2.43738e-02	1.23471e-02	2.45374e-03	1.22648e-03

Table 6. P1 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	1.97729e-03	5.33807e-04	1.34873e-04	5.41901e-06	1.35843e-06
$\Vert u-u_h\Vert_{L^2}$	6.14894e-03	1.61917e-03	3.99372e-04	1.52692e-05	3.81444e-06

Table 7. P2 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	1.40696e-05	1.91059e-06	2.46414e-07	1.95803e-09	2.45484e-10
$\Vert u-u_h\Vert_{L^2}$	5.16536e-05	7.12397e-06	9.13825e-07	7.16198e-09	8.98457e-10

Table 8. P3 convergence analysis
$h$	0.2	0.1	0.05	0.01	0.005
$\Vert p-p_h\Vert_{L^2}$	2.47985e-07	1.81459e-08	1.16742e-09	1.80373e-12	2.89473e-13
$\Vert u-u_h\Vert_{L^2}$	6.13595e-07	4.34515e-08	2.77315e-09	4.19972e-12	1.17702e-12

1.11. Output

The following output example is reproducible using feelpp_toolbox_mixed-poisson-model_2DP2 running on 12 cores with the previous .json and .cfg files on a mesh of typical size $h=0.05$ .

The screenshots are in order : the pressure field, the velocity magnitude field and the velocity field.

pressuremap2

velocitymap2

fluxfield2

1.12. Reference

[BD] Pavel B. Bochev, Clark R. Dohrmann, A computational study of stabilized, low-order $C^0$ finite element approximations of Darcy equations