Examples of 2D Darcy flows

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