Table of content

• Quick summary
• Installation
• How to use

2D Poisson equation

This repository holds an implementation of a solver that solves the 2D-Poisson problem on the unit disc with homogeneous dirichlet boundary conditions. The problem is given by

$$\begin{cases}\nabla^2 u(x, y) = -f(x, y), (x, y)\in\Omega \\ u(x, y) = 0, (x, y) \in \partial \Omega\end{cases}$$

where the domain is the unit disc, $\Omega = {(x, y) : x^2 + y^2 \leq 1}$ with $\partial \Omega = { (x, y) : x^2 + y^2 = 1}$. For a summary of the theory behind the implementation, check out the file 2dPoissonProblem.pdf.

How to install

To install the problem, first make sure that you meet the minimum requirements listed in the file requirements.txt. You should then clone this repository to a designated place on your computer like this. To clone the repository, run these lines in your terminal:

cd navigate/to/your/designated/place
git clone https://github.com/matsiyatzy/Software-and-computing-for-applied-physics.git

How to use

To use this code, follow these steps:

1. Navigate to the place where you installed the program

   cd navigate/to/your/designated/place/Software-and-computing-for-applied-physics

2. Run the solver with this command

   python main.py <num_nodes> <right_hand_side_function> <verbose>

   Where you need to edit <num_nodes>, <right_hand_side_function> and to be like your desired input.

   These values are: - num_nodes: number of nodes (and degrees of freedom) in the finite element mesh of the unit circle, it is given as an integer. - right_hand_side_function: The right hand side of the poisson eqution. Given as an input without spaces. - verbose: Boolean variable with a default value of True. Defines whether or not you want printed outputs during the running of the program:

   Here is an example run:

   python main.py 1000 np.sin(x**2+y**2) True

   And here is another (verbose not provided -> verbose takes the default value True):

   python main.py 10000 -8*np.pi*np.cos(2*np.pi*(x**2+y**2))+16*np.pi**2*(x**2+y**2)*np.sin(2*np.pi*(x**2+y**2))