Elliptic equations

19. Elliptic equations#

19.1. Poisson equation#

Given the volume density source $f (\vec{r})$ and the diffusivity $ν (\vec{r})$ , Poisson equation for the scalar field $ϕ (\vec{r})$ reads

- \nabla \cdot (ν \nabla ϕ) = f \vec{r} \in V

with proper boundary conditions on $\partial V$ . As an example, tipical boundary conditions are:

\begin{array}{r} \begin{aligned} ϕ (\vec{r}) = g (\vec{r}) & \vec{r} \in S_{D} & esserntial - Dirichlet b.c. \\ ν \hat{n} \cdot \nabla ϕ (\vec{r}) = h (\vec{r}) & \vec{r} \in S_{N} & natural - Neumann b.c. \\ a ϕ (\vec{r}) + ν \hat{n} \cdot \nabla ϕ (\vec{r}) = b (\vec{r}) & \vec{r} \in S_{R} & Robin b.c. \end{aligned} \end{array}

19.1.1. Weak formulation#

For $\forall w \in \dots$ (functional space, recall some results about existence and uniqueness of the solution, Lax-Milgram theorem,…)

\begin{array}{r} \begin{aligned} 0 & = \int_{V} w {\nabla \cdot (ν \nabla ϕ) + f} = \\ = \oint_{\partial V} w \hat{n} \cdot (ν \nabla ϕ) + \int_{V} {- ν \nabla \vec{w} \cdot \nabla ϕ + w f} = \end{aligned} \end{array}

Splitting boundary contribution as the sum from single contributions from different regions, and applying boundary conditions, setting $w = 0$ for $\vec{r} \in S_{D}$ (see the ways to prescribe essential boundary conditions),

\begin{array}{r} 0 = \int_{S_{D}} \underset{= 0}{\underset{⏟}{w}} \hat{n} \cdot (ν \nabla ϕ) + \int_{S_{N}} w \underset{= h}{\underset{⏟}{\hat{n} \cdot (ν \nabla ϕ)}} + \int_{S_{R}} w \underset{= b - a ϕ}{\underset{⏟}{\hat{n} \cdot (ν \nabla ϕ)}} + \int_{V} {- ν \nabla \vec{w} \cdot \nabla ϕ + w f} . \end{array}

and rearranging the equation separating terms containing unknowns from known contributions,

\int_{V} ν \nabla w \cdot \nabla ϕ + \int_{S_{R}} w a ϕ = \int_{V} w f + \int_{S_{N}} w h + \int_{S_{R}} w b \forall w \in \dots,

and $ϕ = g$ , for $\vec{r} \in S_{D}$ .

Elliptic equations

Contents

19. Elliptic equations#

19.1. Poisson equation#

19.1.1. Weak formulation#