Poisson Equation

The Poisson equation is a generalisation of the Laplace equation that includes a nonzero source term:

$${\nabla }^{2}u=g$$

where $g$ is a known function of position. In 2D:

$$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=g(x,y)$$

When $g=0$ everywhere, the Poisson equation reduces to the Laplace equation. The Poisson equation arises in problems such as electrostatics (where $g$ is proportional to charge density) and steady-state heat conduction with internal heat generation.

Like the Laplace equation, the Poisson equation has no time variable and requires only Boundary conditions for a unique solution. It can be discretised using the same Laplacian difference equation stencil, with the source term $g$ added to the right-hand side at each grid point. Liebmann’s method can then be applied to solve the resulting linear system iteratively.

Laplace equation | Laplacian difference equation | Liebmann’s method | Boundary conditions