Discrete Poisson Equation - Example

Example

For a 5×5 ( and ) grid with all the boundary nodes prescribed, the system would look like:

\begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} u_{22}, u_{32}, u_{42}, u_{23}, u_{33}, u_{43}, u_{24}, u_{34}, u_{44} \end{bmatrix}^{T}

with

A = \begin{bmatrix} ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 & ~0 \\ -1 & ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 \\ ~0 & -1 & ~4 & ~0 & ~0 & -1 & ~0 & ~0 & ~0 \\ -1 & ~0 & ~0 & ~4 & -1 & ~0 & -1 & ~0 & ~0 \\ ~0 & -1 & ~0 & -1 & ~4 & -1 & ~0 & -1 & ~0 \\ ~0 & ~0 & -1 & ~0 & -1 & ~4 & ~0 & ~0 & -1 \\ ~0 & ~0 & ~0 & -1 & ~0 & ~0 & ~4 & -1 & ~0 \\ ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 & -1 \\ ~0 & ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 \end{bmatrix}

and

b = \begin{bmatrix} -\Delta x^2 g_{22} + u_{12} + u_{21} \\ -\Delta x^2 g_{32} + u_{31} ~~~~~~~~ \\ -\Delta x^2 g_{42} + u_{52} + u_{41} \\ -\Delta x^2 g_{23} + u_{13} ~~~~~~~~ \\ -\Delta x^2 g_{33} ~~~~~~~~~~~~~~~~ \\ -\Delta x^2 g_{43} + u_{53} ~~~~~~~~ \\ -\Delta x^2 g_{24} + u_{14} + u_{25} \\ -\Delta x^2 g_{34} + u_{35} ~~~~~~~~ \\ -\Delta x^2 g_{44} + u_{54} + u_{45} \end{bmatrix}.

As can be seen, the boundary 's are brought to the right-hand-side of the equation. The entire system is 9 × 9 while and are 3 × 3 and given by:

D = \begin{bmatrix} ~4 & -1 & ~0 \\ -1 & ~4 & -1 \\ ~0 & -1 & ~4 \\ \end{bmatrix}

and

-I = \begin{bmatrix} -1 & ~0 & ~0 \\ ~0 & -1 & ~0 \\ ~0 & ~0 & -1 \end{bmatrix}.

Read more about this topic:  Discrete Poisson Equation

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