The classic Poisson equation is one of the most fundamental partial differential equations (PDEs). Although one of the simplest equations, it is a very good model for the process of diffusion and comes up in many applications (for example fluid flow, heat transfer, and chemical transport). It is therefore fundamental to many simulation codes to be able to solve it efficiently and accurately.

This example shows how to up and solve the Poisson equation

$$ d_{ts}\frac{\partial u}{\partial t} + \nabla\cdot(-D\nabla u) = f $$

for a scalar field *u = u(x)* on a unit line. Both the diffusion
coefficient *D* and right hand side source term *f* are assumed
constant and equal to 1. Homogeneous Dirichlet boundary conditions, *u
= 0* are prescribed on all boundaries of the domain. The Poisson
problem is also considered stationary meaning that the time dependent
term can be neglected. The exact solution for this problem is *u(x) =
(-x ^{2}+x)/2*, which can be used to measure the accuracy of
the computed solution.

This model is available as an automated tutorial by selecting **Model
Examples and Tutorials…** > **Classic PDE** > **Poisson Equation**
from the **File** menu, and also as the MATLAB simulation m-script
example ex_poisson1. Step-by-step and
video tutorial instructions, showing how to set up and run this model,
are linked below.