26. Elliptic equations#

26.1. Poisson equation#

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

\[- \nabla \cdot ( \nu \nabla \phi) = f \qquad \vec{r} \in V\]

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

\[\begin{split}\begin{aligned} & \phi(\vec{r}) = g(\vec{r}) && \vec{r} \in S_D && \text{esserntial - Dirichlet b.c.} \\ & \nu \hat{n} \cdot \nabla \phi(\vec{r}) = h(\vec{r}) && \vec{r} \in S_N && \text{natural - Neumann b.c.} \\ & a \phi(\vec{r}) + \nu \hat{n} \cdot \nabla \phi(\vec{r}) = b(\vec{r}) && \vec{r} \in S_R && \text{Robin b.c.} \\ \end{aligned}\end{split}\]
Numerical methods

1-dimensional Poisson equation:

26.1.1. Weak formulation#

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

\[\begin{split}\begin{aligned} 0 & = \int_V w \, \left\{ \nabla \cdot (\nu \nabla \phi) + f \right\} = \\ & = \oint_{\partial V} w \hat{n} \cdot (\nu \nabla \phi) + \int_V \left\{ - \nu \nabla \vec{w} \cdot \nabla \phi + w f \right\} = \\ \end{aligned}\end{split}\]

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),

\[\begin{aligned} 0 = \int_{S_D} \underbrace{w}_{= 0} \hat{n} \cdot (\nu \nabla \phi) + \int_{S_N} w \underbrace{\hat{n} \cdot (\nu \nabla \phi)}_{ = h} + \int_{S_R} w \underbrace{ \hat{n} \cdot (\nu \nabla \phi)}_{ = b - a \phi } + \int_V \left\{ - \nu \nabla \vec{w} \cdot \nabla \phi + w f \right\} \ . \end{aligned}\]

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

\[\int_{V} \nu \nabla w \cdot \nabla \phi + \int_{S_R} w a \phi = \int_{V} w f + \int_{S_N} w h + \int_{S_R} w b \qquad \forall w \in \dots \ ,\]

and \(\phi = g\), for \(\vec{r} \in S_D\).

Different ways to prescribe essential boundary conditions

Strong formulation.

Using Lagrance multiplier - weak formulation of essential boundary conditions. Adding a the essential boundary condition as a constraint with Lagrange multipliers in the weak formulation of the problem,

\[\dots + \int_{S_D} w_D ( \phi - g ) \ ,\]

…

26.1.2. Existence and uniqueness#

Assuming two solutions \(u_1(\mathbf{r})\), \(u_2(\mathbf{r})\) exist for the Poisson problem

\[\begin{split}\begin{cases} - \nabla \cdot ( \nu \nabla u ) = f \qquad \mathbf{r} \in V \\ u|_{S_D} = g \\ \nu \hat{n} \cdot \nabla u|_{S_N} = h \end{cases}\end{split}\]

Their difference \(\delta u (\mathbf{r}) := u_2(\mathbf{r}) - u_1(\mathbf{r})\) then satisfies the homogeneous problem

\[\begin{split}\begin{cases} - \nabla \cdot ( \nu \nabla \delta u ) = 0 \qquad \mathbf{r} \in V \\ \delta u|_{S_D} = 0 \\ \nu \hat{n} \cdot \nabla \delta u|_{S_N} = 0 \end{cases}\end{split}\]

The norm of the gradient of the difference of the solution reads

\[\begin{split}\begin{aligned} \int_{V} \nu |\nabla \delta u|^2 & = \int_{V} \nu \nabla \delta u \cdot \nabla \delta u = \\ & = \int_{V} \nabla \cdot \left( \delta u \, \nu \nabla \delta u \right) - \int_{V} \delta u \, \underbrace{\nabla \cdot ( \nu \nabla \delta u )}_{ = 0 } = \\ & = \oint_{\partial V} \delta u \, \nu \hat{n} \cdot \nabla \delta u = \\ & = \int_{S_D} \underbrace{ \delta u}_{ \delta u|_{S_D} = 0} \nu \hat{n} \cdot \nabla \delta u + \int_{S_N} \delta u \underbrace{\nu \hat{n} \cdot \nabla \delta u}_{\nu \hat{n} \cdot \nabla \delta u|_{S_N} = 0} = \\ & = 0 \ . \end{aligned}\end{split}\]

If \(\nu(\mathbf{r}) > 0\) for \(\forall \mathbf{r} \in V\)1, it follows that

\[\nabla \delta u = \nabla (u_2 - u_1) = 0 \ ,\]

and thus the two solutions differs at most by an additive constant,

\[u_2(\mathbf{r}) - u_1(\mathbf{r}) = c \ .\]

If the Dirichlet boundary has non-null dimension, it forces the value of the functions to coincide on that boundary, \(u_2(\mathbf{r}) = u_1(\mathbf{r})\) on \(S_D\), and thus sets the value of the additive constant \(c\) to be zero, \(c = 0\), and

\[u_2(\mathbf{r}) = u_1(\mathbf{r}) \ ,\]

thus proving the uniqueness of the solution of the Poisson problem, with non-negative diffusion coefficient and non-zero dimension of Dirichlet boundary.

1

As an example, this condition appears in physical systems with non-zero and positive diffusion coefficients in diffusion problems, like thermal conduction or specie diffusion via Fick’s law.