🟢 div-grad#

Here we demonstrate how to use phyem to solve the div-grad problems in different dimensions and different domains.

The general form of the div-grad problem is

- d d^{*} φ^{n} = f^{n},

where $φ^{n}$ and $f^{n}$ are top forms.

2d periodic boundary conditions#

Here we demonstrate how to use phyem, the msepy implementation, to solve the two-dimensional div-grad problem.

In two-dimensions, the mixed formulation of the div-grad problem is

\begin{array}{r} {\begin{aligned} u^{1} & = d^{*} φ^{2}, \\ - d u^{1} & = f^{2} . \end{aligned} \end{array}

We use smooth manufactured solutions for this case. The exact solution for $φ$ is

φ = - \sin (2 π x) \sin (2 π y) .

Exact solutions of $u^{1}$ and $f^{2}$ then follow. We consider the domain to be $Ω = (x, y) \in [0, 1]^{2}$ and it is fully periodic. We use the Multi-crazy domain and mesh for this test. The solver is given below.

div_grad_2d_periodic_manufactured_test(degree, K, c=0)[source]#

Parameters:

degreeint
The degree of the mimetic spectral elements.

Kint
In total we will use $4 * K * K$ elements.

cfloat, default=0
The deformation factor of the Multi-crazy domain and mesh.

Returns:

phi_error: float
The $L^{2}$ -error of solution $φ_{h}^{2}$ .

u_error: float
The $L^{2}$ -error of solution $u_{h}^{1}$ .

Examples#

Below, we use mimetic spectral elements of degree 2 on a uniform mesh of $4 * 4 * 4$ $(K = 4)$ elements.

>>> errors4 = div_grad_2d_periodic_manufactured_test(2, 4)
>>> float(errors4[0])  
0.01...

We increase $K$ to $K = 8$ , we do

>>> errors8 = div_grad_2d_periodic_manufactured_test(2, 8)

We can compute the convergence rate of the $L^{2}$ -error of solution $φ_{h}^{2}$ by

>>> import numpy as np
>>> rate = (np.log10(errors4[0]) - np.log10(errors8[0])) / (np.log10(1/4) - np.log10(1/8))
>>> float(round(rate, 1))
2.0

The optimal convergence rate is obtained.

2d general boundary conditions#

Here we repeat the test, but with essential boundary $tr u^{1}$ on faces $y = 0$ and $y = 1$ , and natural boundary condition $tr (⋆ φ^{2})$ on faces $x = 0$ and $x = 1$ .

The implementation is

div_grad_2d_general_bc_manufactured_test(degree, K, c=0.0)[source]#

Parameters:

degreeint
The degree of the mimetic spectral elements.

Kint
In total we will use $K * K$ elements.

cfloat, default=0
The deformation factor of the Crazy domain and mesh.

Returns:

phi_error: float
The $L^{2}$ -error of solution $φ_{h}^{2}$ .

u_error: float
The $L^{2}$ -error of solution $u_{h}^{1}$ .

Examples#

If we solve it with $4 \times 4$ elements (note that here we use a different mesh compared to the periodic test) at polynomial degree 2,

>>> errors4 = div_grad_2d_general_bc_manufactured_test(2, 4)
>>> float(errors4[0])  
0.06...

We increase $K$ to $K = 8$ , we do

>>> errors8 = div_grad_2d_general_bc_manufactured_test(2, 8)

We can compute the convergence rate of the $L^{2}$ -error of solution $φ_{h}^{2}$ by

>>> import numpy as np
>>> rate = (np.log10(errors4[0]) - np.log10(errors8[0])) / (np.log10(1/4) - np.log10(1/8))
>>> float(round(rate, 1))
2.0

Again, the optimal convergence rate is obtained.

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🟢 div-grad

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🟢 div-grad#

2d periodic boundary conditions#

Examples#

2d general boundary conditions#

Examples#