For those involved in science in the broad sense, it is important at some stage to solve Partial Differential Equations (PDE). When geometry of the problem is complex, it is often impossible to solve the equation analytically. In such cases, numerical methods must be used. The method that has been widely used for this type of problem in recent years is Finite Element Method (FEM). Many commercial software environments have been developed that use FEM for this type of application: ANSYS, ADINA, COMSOL, etc. However, these are commercial programs, sometimes quite expensive. At this point, it is worth mentioning a number of free, open-source programs that offer the same capabilities. One such programming environment that has seen dynamic growth in recent years is FEniCS library. Open-source programs often do not look as nice as commercial programs with their beautiful interfaces. However, there is one significant advantage: we have more control over what we do, although the threshold for entering given environment can be high.
With this post, I would like to start series of posts on using FEniCS to solve PDEs. It is worth mentioning that FEniCS is developed in scientific institutions and, in a sense, has full approval for use in scientific publications.
FEniCS logo by Mattias Schläger, CC BY-SA 4.0
I won't bore you with the details of how FEM works. Unless someone is interested, of course – I'd be happy to explain how it works from the inside out. I'll focus strictly on using FEniCS, treating it as a tool for solving PDEs that describe vibrations, motion of electric charges, heat flow, etc. This post is introduction to the topic; the next one will be about preparing working environment. Working in FEniCS – as open-source environment – is more convenient in Linux. However, thanks to dockers, it is now possible to set up convenient environment in Windows with minimal effort. The next post will cover how to create a working environment in Windows.
Example solution of the Poisson equation.
Photos:
[1] — FEniCS logo by Mattias Schläger
[2] — Solution of some Poisson's equation
