Feel++

Mixed Poisson

Notations and units

Notation

Quantity

\$k\$

conductivity

\$\Lambda\$

resistivity

Equations

Mixed Poisson equations are

\$\begin{align} \boldsymbol{u} + k\nabla p &= \boldsymbol{f}\\ \nabla\cdot \boldsymbol{u} &= g \end{align}\$

completed by boundary conditions.

The conductivity \$k\$ can be non-linear. If it depends only on the primal variable \$p\$, it is already handle by the model, in other cases, one needs to provide the corresponding expression.

MixedPoisson Toolbox

The model is described in a json file which path is given by the option mixedpoisson.model_json. The construction of this file is detailed in the following sections.

Models

The models are not considered for now.

Model section
"Model": "HDG"

Materials

The definition of the conductivity \$k\$ depends on the material, it can be linear or non-linear. In the linear case, it is given in the material we work on by the keyword cond and in the non-linear case, by condNL.

Material section
"Materials":
{
    "<marker>":
    {
        "name": "copper",
        "cond": "1",
	"condNL": "2*p:p"
    }
}

The keywords cond and condNL can be changed respectively by the options mixedpoisson.conductivity_json and mixedpoisson.conductivityNL_json.

Boundary Conditions

All boundary conditions are described in the same way

Listing : boundary conditions in json
"BoundaryConditions":
{
    "<field>":
    {
        "<bc_type>":
        {
            "<marker>":
            {
                "<option1>":"<value1>",
                "<option2>":"<value2>",
                // ...
            }
        }
    }
}

Different types of boundary condition are available.

Dirichlet condition
\$p = g_D\$

Field

Type

Option

Value

potential

Dirichlet

expr

\$g_D\$
Neumann condition
\$-k\nabla p \cdot\boldsymbol{n} = g_N\$

Field

Type

Option

Value

potential

Neumann

expr

\$g_DN\$ or \$-k\nabla p\$

The choice between \$g_DN\$ or \$-k\nabla p\$ is base on the dimension of the expression.

Robin condition
\$-k\nabla p \cdot\boldsymbol{n} + g_R^1 p = g_R^2\$

Field

Type

Option

Value

potential

Robin

expr1

\$g_R^1\$

potential

Robin

expr2

\$g_R^2\$
Integral boundary condition
\$\int_{\Gamma_I} \boldsymbol{u}\cdot\boldsymbol{n} = g_I\$

Field

Type

Option

Value

flux

Integral

expr

\$g_I\$

Source Terms

The source terms \$f\$ and \$\boldsymbol{g}\$ are treated as boundary condtions.

Field

Type

Option

Value

potential

SourceTerm

expr

\$f\$

flux

SourceTerm

expr

\$\boldsymbol{g}\$

Post Process

Two fields can be exported, the potential \$p\$ and the flux \$\boldsymbol{u}\$.

Post Process section
"PostProcess":
{
    "Fields":["potential","flux"]
}

Create applications

In order to solve linear problem, an application should contain at least

Minimal Linear case
    typedef FeelModels::MixedPoisson<FEELPP_DIM,FEELPP_ORDER> mp_type;
    auto MP = mp_type::New("mixedpoisson");
    MP->init();
    MP->assembleAll();
    MP->solve();
    MP->exportResults();