Coefficient Form PDEs

Coefficient Form PDEs (CFPDEs) is an application of software Feelpp (website). Feelpp is a solver of differential equation which use finite elements.

Coefficient Form PDEs can solve many types of partial differential equations derived from diffusion/convection model. This application is highly adaptive, we can implement lot of physic (elastic, heat, Maxwell equation in cartesian, axisymmetry, in static or transient…​) and give acces of last features (postproccess, new type of elements…​).

1. Differential Formulation

Coefficient Form PDEs solves two equations of the form : the Scalar Equation (of scalar unknow) and Vectorial Equation (of vectorial unknow).

1.1. Scalar Form

Differential Formulation of cfpdes : Scalar Form

Set the domain of resolution \(\Omega \subset \mathbb{R}^d\)

\[d \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u - \alpha u + \mathbf{γ} \right) + \mathbf{β} \cdot \nabla u + a u = f\]

With :

Symbol Description Dimension

  • \(u\)

scalar unknown

\(1\)

  • \(d\)

damping or mass coefficient

\(1\)

  • \(c\)

diffusion coefficient

\(1\) or \(m \times m\)

  • \(\alpha\)

conservative flux convection coefficient

\(m\)

  • \(\mathbf{γ}\)

conservative flux source term

\(m\)

  • \(\mathbf{β}\)

convection coefficient

\(m\)

  • \(a\)

absorption or reaction coefficient

\(1\)

  • \(f\)

source term

\(1\)

1.2. Vectorial Form

Differential Formulation of cfpdes : Vectorial Form

Set the domain of resolution \(\Omega \subset \mathbb{R}^d\) and unknow \(\mathbf{u} : \Omega \rightarrow \mathbf{R}^n\)

\[d \frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot \left( -c \nabla \mathbf{u} + \mathbf{γ} \right) + \mathbf{β} \cdot \nabla \mathbf{u} + a \mathbf{u} = \mathbf{f} \text{ on domain } \Omega \in \mathbb{R}^d\]

With :

Symbol Description Dimension

  • \(\mathbf{u}\)

scalar unknown

\(n\)

  • \(d\)

damping or mass coefficient

\(1\)

  • \(c\)

diffusion coefficient

\(1\) or \(m \times m\)

  • \(\mathbf{γ}\)

conservative flux source term

\(m \times m\)

  • \(\mathbf{β}\)

convection coefficient

\(m\)

  • \(a\)

absorption or reaction coefficient

\(1\)

  • \(f\)

source term

\(n\)
It is possible to couple many equations of this type (either scalar or vectorial).

2. Weak Formulation

Now, we see the weak formulation of two type of equations solve by CFPDEs.

2.1. Scalar Form

We multiply Differential Formulation of cfpdes : Scalar Form with \(\mathbf{\phi} \in H^1(\Omega)\) and integrate over \(\Omega\) :

\[\small{ \int_{\Omega}{ d \frac{\partial u}{\partial t} \, \phi \, dx dy dz } + \int_{\Omega}{ \left( \nabla \cdot \left( -c \nabla u - \alpha u + \mathbf{γ} \right) \right) \, \phi \, \, dx dy dz } + \int_{\Omega}{ \left( \mathbf{β} \cdot \nabla u \right) \, \phi \, \, dx dy dz } + \int_{\Omega} { a \; u \, \, dx dy dz } = \int_{\Omega}{ f \, \phi \, \, dx dy dz } }\]

With Formula of Green (First Form), we have :

\[\small{ \begin{eqnarray*} \int_{\Omega}{ \left( \nabla \cdot \left( c \, \nabla u + \alpha \, u - \gamma \right) \right) \, \phi \ dx dy dz } = \int_{\Omega}{ \left( c \, \nabla u + \alpha \, u - \gamma \right) \, \nabla \phi \ dx dy dz } + \int_{\Gamma}{ \left( ( c \nabla u + \alpha u + \gamma ) \cdot \mathbf{n} \right) \, \phi \ d \Gamma } \end{eqnarray*} }\]

We impose the boundary conditions of Dirichlet, Neumann and Robin on the boundary \(\partial \Omega = \Gamma = \Gamma_N \cup \Gamma_R\) with the term :

  • Strong Dirichlet : \(u = u_0\) on \(\Gamma_D\)

  • Neumann condition : \(\int_{\Gamma_N}{ g_N \, \phi \, dx dy dz }\) on second term

  • Robin condition : \(\int_{\Gamma_R}{ h_R u \, \phi \, dx dy dz }\) on first term and \(\int_{\Gamma_R}{ g_R \, \phi \, dx dy dz }\) on second term

Weak Formulation of cfpdes : Scalar Form
\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ d \, \frac{du}{dt} \, \phi \ dx dy dz } + \int_{\Omega}{ \left( c \nabla u + \alpha u - \gamma \right) \cdot \nabla \phi \ dx dy dz} + \int_{\Omega}{ \left( \mathbf{β} \cdot \nabla u \right) \, \phi \ dx dy dz } + \int_{\Omega}{ a \, u \, \phi \ dx dy dz } + \int_{\Gamma_R}{ h_R u \, \phi \, dx dy dz } \\ = \int_{\Omega}{ f \, \phi \ dx dy dz} + \int_{\Gamma_N}{ g_N \, \phi \, dx dy dz } + \int_{\Gamma_R}{ g_R \, \phi \, dx dy dz } \end{eqnarray*} }\]

2.2. Vectorial Form

We multiply Differential Formulation of cfpdes : Vectorial Form with \(\mathbf{\phi} \in H^1(\Omega)\) and integrate over \(\Omega\) :

\[\small{ \int_{\Omega}{ d \frac{\partial \mathbf{u}}{\partial t} \, \phi \, dx dy dz } + \int_{\Omega}{ \left( \nabla \cdot \left( - c \nabla \mathbf{u} + \mathbf{γ} \right) \right) \, \phi \, \, dx dy dz } + \int_{\Omega}{ \left( \mathbf{β} \cdot \nabla \mathbf{u} \right) \, \phi \, \, dx dy dz } + \int_{\Omega} { a \; \mathbf{u} \, \, dx dy dz } = \int_{\Omega}{ \mathbf{f} \, \phi \, \, dx dy dz } }\]

With Formula of Green (First Form), we have :

\[\small{ \begin{eqnarray*} \int_{\Omega}{ \left( \nabla \cdot \left( c \, \nabla \mathbf{u} - \gamma \right) \right) \, \phi \ dx dy dz } = \int_{\Omega}{ \left( c \, \nabla \mathbf{u} - \gamma \right) \, \nabla \phi \ dx dy dz } + \int_{\Gamma}{ \left( ( c \nabla \mathbf{u} + \gamma ) \cdot \mathbf{n} \right) \, \phi \ d \Gamma } \end{eqnarray*} }\]

We impose the boundary conditions of Strong Dirichlet, Neumann and Robin on the boundary \(\partial \Omega = \Gamma = \Gamma_N \cup \Gamma_R\) with the term :

  • Strong Dirichlet : \(u = u_0\) on \(\Gamma_D\)

  • Neumann condition : \(\int_{\Gamma_N}{ \mathbf{g_N} \, \phi \, dx dy dz }\) on second term

  • Robin condition : \(\int_{\Gamma_R}{ h_R \mathbf{u} \, \phi \, dx dy dz }\) on first term and \(\int_{\Gamma_R}{ \mathbf{g_R} \, \phi \, dx dy dz }\) on second term

Weak Formulation of cfpdes : Vectorial Form
\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ d \, \frac{d\mathbf{u}}{dt} \, \phi \ dx dy dz } + \int_{\Omega}{ \left( c \nabla \mathbf{u} - \gamma \right) \cdot \nabla \phi \ dx dy dz} + \int_{\Omega}{ \left( \mathbf{β} \cdot \nabla \mathbf{u} \right) \, \phi \ dx dy dz } + \int_{\Omega}{ a \, \mathbf{u} \, \phi \ dx dy dz } + \int_{\Gamma_R}{ h_R \mathbf{u} \, \phi \, dx dy dz } \\ = \int_{\Omega}{ f \, \phi \ dx dy dz} + \int_{\Gamma_N}{ \mathbf{g_N} \, \phi \, dx dy dz } + \int_{\Gamma_R}{ \mathbf{g_R} \, \phi \, dx dy dz } \end{eqnarray*} }\]

3. Specific case for axisymmetric geometry