5.3 Boundary Value Problems: Elliptic DE

Standard problem: two-dimensional potential equation,

$$\frac{\partial^{2} u}{\partial x^{2}} +\frac{\partial^{2} u}{\partial y^{2}} = -\rho(x,y)$$

For general $\rho(x,y)$ this is Poisson's equation; if $\rho \equiv 0$ it is called Laplace's equation.

Assuming $\Delta y = \Delta x \equiv \Delta l$ we have

$$\frac{1}{(\Delta l)^{2}} \left[ \delta_{i}^{2} u_{i,j} +\delta_{j}^{2} u_{i,j} \right] = -\rho_{i,j}$$

or

$$\frac{1}{(\Delta l)^{2}} \left[ u_{i+1,j} \right. - \left. 2u_{i,j}+ u_{i-1,j} +u_{i,j+1}-2u_{i,j}+u_{i,j-1} \right] = - \rho_{i,j}$$

$$\hspace{6em}(i=1,2,\dots N;\;\; j=1,2,\dots M)$$

Construct a vector $\mbox{$\bf v$}$ of length $N.M$ by linking together the rows of the matrix $\{ u_{i,j}\}$:

$$v_{r} = u_{i,j}\,,\;\;\;\;{\rm with}\;\; r=(i-1)M+j$$

The potential equation then reads

$$v_{r-M}+v_{r-1}-4v_{r}+v_{r+1}+v_{r+M}= - (\Delta l)^{2} \rho_{r}$$

or

$$\mbox{${\bf A}$} \cdot \mbox{$\bf v$} = \mbox{$\bf b$}$$

with $\mbox{$\bf b$} \equiv - (\Delta l )^{2} \{ \rho_{1}, \dots \rho_{N.M}\}^{T}$ and

$$\mbox{${\bf A}$} \equiv \left( \begin{array}{cccccc} -4 & 1 & \dots & 1 & & \\ 1 &-4 & 1... ...dots &\ddots &\ddots & & \\ 1 & & & & & \\ &\ddots & & & & \end{array}\right)$$

Treating the boundaries:

Assume $u_{i,j}=u_{i,j}^{0}$ to be given along the sides: $\Longrightarrow$ $v_{1}=u_{1,1}^{0}$ etc.

At the interior points we have

$$-4v_{7}+v_{8}+v_{12} = - (\Delta l)^{2} \rho_{2,2}-u_{2,1}^{0}-u_{1,2}^{0}$$

etc., yielding a modified system matrix $\mbox{${\bf A}$}$ and vector $\mbox{$\bf b$}$.
More specifically, the matrix $\mbox{${\bf A}$}$ has the form

Figure 5.4: Treatment of Dirichlet-type boundary conditions $u_{i,j}=$ $u_{i,j}^{0}$ in the case of a $5 \times 5$ lattice

The vector $\mbox{$\bf v$}$ consists of the nine elements $v_{7},v_{8},v_{9},$ $v_{12},v_{13},v_{14},$ $v_{17},$ $v_{18},$ $v_{19}$, and the vector $\mbox{$\bf b$}$ has components

$$b_{7} = - (\Delta l)^{2} \rho_{7}-u_{1,2}^{0} - u_{2,1}^{0}$$

$$b_{8} = - (\Delta l)^{2} \rho_{8}-u_{1,3}^{0}$$

$$b_{9} = - (\Delta l)^{2} \rho_{9}-u_{1,4}^{0} - u_{2,5}^{0}$$

$$b_{12} = - (\Delta l)^{2} \rho_{12}-u_{3,1}^{0}$$

$$b_{13} = - (\Delta l)^{2} \rho_{13}$$

$$b_{14} = - (\Delta l)^{2} \rho_{14}-u_{3,5}^{0}$$

$$b_{17} = - (\Delta l)^{2} \rho_{17}-u_{4,1}^{0} - u_{5,2}^{0}$$

$$b_{18} = - (\Delta l)^{2} \rho_{18}-u_{5,3}^{0}$$

$$b_{19} = - (\Delta l)^{2} \rho_{19}-u_{4,5}^{0} - u_{5,4}^{0} \nonumber$$

Figure 5.5: Potential equation on a $5 \times 5$ lattice: at the points $\circ$ the values of the potential $u(x,y)$ are given (Dirichlet boundary conditions)

Neumann boundary conditions of the form

$$\left( \frac{\partial u}{\partial x} \right)_{i,j} = \alpha_... ...ft( \frac{\partial u}{\partial y} \right)_{i,j} = \beta_{i,j}$$

may be treated using a linear approximation for $u(x,y)$ at the rim. Again considering the previous example, we proceed as follows:

• Add an additional ``skin layer''of lattice points points with linearly extrapolated vales of $u$:
  

  $$u_{0,1} = u_{2,1}- 2\alpha_{1,1} \, \Delta l$$

  $$u_{0,2} = u_{2,2}- 2\alpha_{1,2} \, \Delta l$$

  $$\vdots$$

  $$u_{1,0} = u_{1,2}- 2\beta_{1,1} \, \Delta l$$

  $$\vdots \nonumber$$

  
  

• At the original boundary points, such as $(1,1)$, we have
  
  $$u_{2,1}-2u_{1,1}+u_{0,1}+u_{1,2}-2u_{1,1}+u_{1,0}=-\rho_{1,1} (\Delta l)^{2}$$
  
  
  Expressing the external values by the internal ones we have
  
  $$u_{2,1}-2u_{1,1}+u_{2,1}+u_{1,2}-2u_{1,1}+u_{1,2}=-\rho_{1,1... ... l)^{2} +2\alpha_{1,1} \, \Delta l + 2\beta_{1,1} \, \Delta l$$
  
  

$\Longrightarrow$Same form of the discretized Poisson equation as in the interior region but with ``effective'' charge densities.

Putting things together, we find

$$\left( \begin{array}{ccccccccccccccc} -4& 2& & & & \vert& 2... ...& \vert& & & & & & \vert& & &\\ \end{array}\right) \nonumber$$

Treatment of Neumann-type boundary conditions in the case of a $5 \times 5$ lattice