Liebmann’s Method

Liebmann’s method (also known as Gauss-Seidel iteration applied to PDEs) is an iterative solver for the 2D Laplace equation, based on the Laplacian difference equation.

Update Rule

At each interior grid point $(i,j)$, apply the averaging update:

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

Algorithm

1. Initialise $u_{i,j}$ (e.g. to zero or an interpolation of the Boundary conditions).
2. Apply the update rule to every interior point.
3. Check convergence: if ${\max }_{i,j}|u_{i,j}^{\text{new}}-u_{i,j}^{\text{old}}|<\epsilon$, stop.
4. Otherwise, set $u^{\text{old}}\leftarrow u^{\text{new}}$ and repeat.

Convergence

Liebmann’s method converges because the matrix associated with the Laplacian difference equation is diagonally dominant. Diagonal dominance guarantees that the Gauss-Seidel iteration converges to the unique solution.

Laplace equation | Laplacian difference equation | Diagonal dominance | Boundary conditions