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Project: Approximation of PDEs

Developed by Patrik Rác

Description

Implementation of a 2D finite element solver for the cours MU4MA029. The code is structured in to the following files:

  • main.py - main file, contains the main function and the convergence benchmark.
  • geometry.py - contains all methods related to the mesh generation.
  • assembly.py - contains all methods related to the assembly of the local and global matrices as well as the right-hand side vector.
  • utilites.py - contains other usefull methods related to visualization and the computation of the boundary elements or mesh refinement.
  • tests.py - contains unit tests for the mass matrix and stiffness matrix.

The parameters concerning the funciton f are defined in the main() method of the main.py file and have to be changed in the source code if deemed nessecarry.
By default the program sets the paramters to the values p=3 and q=3, as-well as the predefined values for b = [1., 1.] and c=1..

Additionally, a precomputed convergence plot is provided in the file convergence.png, showing the quadratic convergence of the method.

Requirements

The program is written in Python 3.10 and relies on libraries written for scientific computing, namely NumPy, SciPy and Matplotlib.
Furthermore, it uses the unittest library for testing.

Usage

The program can be executed by running the following command in the terminal:
python3 main.py [N Nl] [--bench]

The optional arguments are:

  • N - number of subdivisions per unit length
  • Nl - parameter degfining l=Nl*h where h is the length of the subdivision
  • --bench - if present, the program will run the convergence benchmark. (This might take a while...)

By defaul (calling the program without parameters) the program will use N=40 and Nl=20. It is nessecarry that N is divisible by Nl, in order for the problem to be well-defined.

Examples

python3 main.py 100 50
python3 main.py 100 20
python3 main.py --bench
python3 main.py 100 50 --bench

Testing

Additionally, test on the mass matrix and stiffness matrix can be run by executing the following command in the terminal:
python3 -m unittest tests.py [-v]
Where the optional argument -v activates verbosity.

Examples

python3 -m unittest tests.py
python3 -m unittest tests.py -v