Motivation#
The first goal of the workshop is to approximate
on the interval \([0, 1]\).
We want to start with a simple example to illustrate the basic idea behind the finite element method, which is sub-dividing the domain of interest \(\Omega\) into smaller sub-domains, called elements, and on each element represent a function \(u_h\) by a linear combination of simple basis functions, e.g. polynomials.
Certain problems can be solved analytically, but only for specific boundary conditions, material properties, and geometries.
Selection of approximation functions#
There are many different choices we could choose for approximating \(g\):
Global polynomial approximation (e.g. piecewise linear, quadratic, cubic, etc.)
Finite Fourier series
Piecewise polynomial approximation (e.g. Taylor series)
When solving PDEs, we will encounter singularities and non-smooth solutions (e.g. kinks). Both these features make global polynomial approximation and Fourier series less attractive.

We could then increase the number of elements used

Motivating example: Heat equation with different materials#
This can for instance be seen in heat transfer equation between different materials. We define a domain \(\Omega\in \mathbb{R}^d\) as the union of two disjoint domains \(\Omega_0\) and \(\Omega_1\)
Material#
\(c\) is the coefficient of thermal diffusivity
which is discontinuous at \(\Gamma\), while the governing partial differential equation yields