Poisson Equation

Model Data

Type: classic pde

Physics Modes: poisson equation

Keywords: 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 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) = (-x2+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.