# Poisson Equation with a Point Source

## Model Data

**Type:**
quickstart
classic pde

**Physics Modes:**
poisson equation

**Keywords:**
point source
validation

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 again and again in many applications such as in fluid flow, heat transfer, and chemical transport.

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 circle with radius *r = 1*. The
diffusion coefficient *D = 1* and right hand side source term *f =
δ(0)* which prescribes a point source at the center. The Poisson
problem is also considered stationary meaning that time dependent term
can be neglected. With these assumptions the equation simplifies to

$$ - \Delta u = \delta(0,0) $$

Moreover, homogeneous Dirichlet boundary conditions are prescribed on all
boundaries of the domain, that is *u = 0* on the boundary. The exact solution
for this problem is *u(x,y) = -1/(2π)log( r )* and 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…** > **Quickstart** > **Poisson Equation with
a Point Source** from the **File** menu. Or alternatively, follow the
linked step-by-step instructions.