Step0.md

Step-0

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However, you need to exit Raw mode to display the Jupyter Notebook correctly.

In [the Step-0 Jupyter Notebook], I provide a basic introduction.

We solve the following diffusion + reaction equation, with a source term:

$$ -\frac{d}{dx}D\frac{du}{dx} + \sigma u = q \qquad \forall x \in [0,L]$$

We will assume all material properties $D, \sigma,q$ to be constant for now.

We use the following boundary conditions:

• at $x=0$, we impose a value $ u(0)=u_{\text{left}}$
• at $x=L$, we impose a value $ u(L)=u_{\text{right}}$

These conditions are known as Dirichlet boundary conditions.

Meshing the domain

The interval $[0,L]$ is meshed using $N$ cells. In FEM, meshcells are known as 'elements'. The vertices of the mesh are denoted by $x_i$ and there are $N+1$ vertices. Obviously, $x_0=0$ and $x_N=L$.

Choice of solution space

We will approximate the solution to the above problem as a piece-wise linear function. Between two consecutive vertices, the function is linear, as shown in the plot below.

Many FEM concepts such as the assembly of the global matrix per element, the reference element, ... are not discussed at first.

At the end of the Step-0 Jupyter Notebook, I do the global assembly by looping over elements.