Laplacian Difference Equation

The Laplacian difference equation is the finite difference form of the 2D Laplace equation, obtained by replacing both second derivatives with centred Finite differences.

$$u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}=0$$

Rearranging:

$$u_{i,j}=\frac{1}{4}\left(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}\right)$$

This says that at each interior grid point, the value of $u$ equals the average of its four neighbours. This averaging structure is the basis of Liebmann’s method.

The equation must be satisfied at every interior grid point simultaneously. Together with the Boundary conditions, this gives a large linear system. Rather than solving it directly, Liebmann’s method solves it iteratively by repeatedly applying the averaging rule until convergence.

Laplace equation | Liebmann’s method | Finite differences | Boundary conditions | Diagonal dominance