2 or more Dimensions

Poisson's and Laplace's equations can be solved in 2 or more dimensions by simple generalisations of the schemes discussed for 1D. However the resulting matrix will not in general be tridiagonal. The discretised form of the equation takes the form

$$V_{m,n-1} + V_{m,n+1} + V_{m-1,n} + V_{m+1,n} - 4 V_{m,n} = \delta x^2\cdot f_{m,n}.$$ (3.8)

The 2 dimensional index pairs $\{m,n\}$ may be mapped on to one dimension for the purpose of setting up the matrix. A common choice is so-called dictionary order,

$$\begin{eqnarray*} (1,1), (1,2), \ldots (1,N), (2,1), (2,2), \ldots (2,N), \ldots (N,1), (N,2), \ldots (N,N) \end{eqnarray*}$$

Alternatively Fourier transformation can be used either for all dimensions or to reduce the problem to tridiagonal form.