Jacobi Algorithm for solving Laplace's Equation
It is thus fairly straightforward to extend this equation to the
three-dimensional case. Whether we solve Eq.
(17)
or Eq.
(18), the solution strategy remains the same.
We know the values of \( u \) at \( i=0 \) or \( i=n+1 \) and at \( j=0 \) or
\( j=n+1 \) but we cannot start at one of the boundaries and work our way into and
across the system since Eq.
(17) requires the knowledge
of \( u \) at all of the neighbouring points in order to calculate \( u \) at any
given point.