Discretization of Poisson's equation with 2d permittivity tensor

Question

I have to discretize a generalized Poisson equation in 2D which is

$$\nabla\cdot(\varepsilon \nabla \phi )=-\rho$$

My problem is that here $\varepsilon$ is $2\times2$ permittivity tensor

where

$$\varepsilon = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} \\ \varepsilon_{yx} & \varepsilon_{yy} \end{bmatrix}$$

Can anyone help me to discretize this Poisson equation using the finite difference method?

Answer 1

You have \begin{align} \nabla \cdot \left(\begin{pmatrix} a & b \\ c & d\end{pmatrix} \nabla \phi \right) &= \nabla\cdot\begin{pmatrix} a \partial_x\phi + b\partial_{y}\phi \\ c\partial_{x}\phi+d\partial_y \phi\end{pmatrix} \\ &= \partial_x(a\partial_x\phi) + \partial_x(b\partial_y\phi) + \partial_y(c\partial\phi_{x})+\partial_y(d\partial_y\phi). \end{align}

You can try and apply central finite differences with step one half for the non-mixed terms: \begin{align} \partial_x f(x,y) &=\frac{f(x+\frac{1}{2}h_x,y)-f(x-\frac{1}{2}h_x,y)}{h_x} + O(h_x^2), \\ \partial_y f(x,y) &=\frac{f(x,y+\frac{1}{2}h_y)-f(x,y-\frac{1}{2}h_y)}{h_y} + O(h_y^2) \end{align}

For simplicity consider $h_x = h_y = 1$, then: \begin{align} \partial_{x}(a\partial_x\phi) &\approx a_{i+\frac{1}{2},j}(\partial_x\phi)_{i+\frac{1}{2},j}-a_{i-\frac{1}{2},j}(\partial_x\phi)_{i-\frac{1}{2},j} \\ &\approx a_{i+\frac{1}{2},j}(\phi_{i+\frac{1}{2},j}-\phi_{i-\frac{1}{2},j})_{i+\frac{1}{2},j} - a_{i+\frac{1}{2},j}(\phi_{i+\frac{1}{2},j}-\phi_{i-\frac{1}{2},j})_{i-\frac{1}{2},j} \\ &= a_{i+\frac{1}{2},j}(\phi_{i+1,j}-\phi_{i,j})-a_{i-\frac{1}{2},j}(\phi_{i,j}-\phi_{i-1,j}) \\ &= a_{i+\frac{1}{2},j}\phi_{i+1,j} - (a_{i+\frac{1}{2},j}+a_{i-\frac{1}{2},j})\phi_{i,j} + a_{i-\frac{1}{2},j}\phi_{i-1,j} \\ &\approx \frac{a_{i+1,j}+a_{i,j}}{2}\phi_{i+1,j} - \frac{a_{i+1,j}+2a_{i,j}+a_{i-1,j}}{2}\phi_{i,j} + \frac{a_{i,j}+a_{i-1,j}}{2}\phi_{i-1,j}. \end{align}

For the mixed terms you can apply central finite differences with a full step: \begin{align} \partial_{x}(b\partial_y\phi) &\approx \frac{1}{2}b_{i+1,j}(\partial_y\phi)_{i+1,j} - \frac{1}{2}b_{i-1,j}(\partial_y\phi)_{i-1,j} \\ &\approx \frac{1}{4}b_{i+1,j}(\phi_{i,j+1}-\phi_{i,j-1})_{i+1,j} - \frac{1}{4}b_{i-1,j}(\phi_{i,j+1}-\phi_{i,j-1})_{i-1,j} \\ &= \frac{1}{4}b_{i+1,j}\phi_{i+1,j+1}-\frac{1}{4}b_{i+1,j}\phi_{i+1,j-1} - \frac{1}{4}b_{i-1,j}\phi_{i-1,j+1}+\frac{1}{4}b_{i-1,j}\phi_{i-1,j-1}. \end{align}

You can do a similar thing for the remaining terms. For some other discretisations you can have a look at this paper and the references within.