# Poisson Equation

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 very fundamental to many simulation codes to be able to solve it correctly and efficiently.

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. The Poisson problem is also considered
stationary meaning the time dependent term can be neglected.
Homogeneous Dirichlet boundary conditions, *u = 0* are prescribed on
all boundaries of the domain. 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. Or alternatively, follow the linked step-by-step instructions.