The goal of this guide is to present the theory of the finite element method using simple examples, without excessive mathematical rigor. The material is aimed at an engineer with a knowledge of the basics of higher mathematics.
Partial differential equations have long been successfully solved both by analytical methods — for simple geometries — and by numerical ones — for complex geometries. In practice geometries are usually complex, so one has to solve numerically. Among numerical methods, the most widespread are the finite difference method (FDM) and the finite element method (FEM). The finite element method is preferable because it allows a nonuniform mesh of simplices, which makes it possible to concentrate computational resources where they are really needed.
This guide deals with one of the simplest equations — the nonstationary heat conduction equation. In practice you will most likely face more complex problems, but once you master the method presented here, more general cases will become much clearer. You are strongly encouraged to verify all formulas yourself.