This repository contains the work I completed during my research scholarship at the University of Newcastle in 2023-2024. The project focused on implementing and exploring numerical methods, specifically the Virtual Element Method (VEM), and contributions in both Julia and MATLAB.
This project solves the possion equation:
$ \begin{cases}
-\Delta{u} & = f \quad \text{in} \ \Omega
u & = g \quad \text{on}\ \delta\Omega
\end{cases} $
Where $f,g \in L_2$
The primary goal of this research project was to explore and implement numerical methods, specifically focusing on the Virtual Element Method (VEM). This repository includes:
Here’s a complete list of the files in this repository:
| File | Description |
|———————————–|——————————————————————————|
| .gitignore | Specifies files ignored by Git. |
| Project.toml | Julia project dependencies. |
| README.md | This README file describing the repository structure. |
| File | Description |
|———————————–|——————————————————————————|
| LaTeX/README.md | Documentation related to the LaTeX report. |
| LaTeX/appendix.tex | Appendix with MATLAB and Julia code listings. |
| LaTeX/diagram.tex | Diagrams used in the LaTeX report. |
| LaTeX/jlcode.sty | Custom Julia syntax highlighting for LaTeX. |
| LaTeX/main.tex | Main LaTeX document for the report. |
| LaTeX/sources.bib | Bibliography file for references. |
| LaTeX/vem-preamble.sty | Preamble for the LaTeX report including packages and settings. |
| File | Description |
|———————————–|——————————————————————————|
| MATLAB/PolygonTests.m | MATLAB file containing tests for polygon meshes in the VEM context. |
| File | Description |
|———————————–|——————————————————————————|
| src/L_domain.jl | Julia code for handling the L-shaped domain problem. |
| src/plot_solution.jl | Julia script to plot the solution to the VEM problem. |
| src/read_mat.jl | Julia script for reading .mat files. |
| src/square_domain.jl | Julia code for handling a square domain. |
| src/vem_method.jl | Julia implementation of the VEM method. |
| src/vem_problem.jl | Julia code defining the VEM problem struct. |
| src/vem_solver.jl | Julia code to solve the VEM problem. |
| File | Description |
|———————————–|——————————————————————————|
| check/debugging/output.txt | Debug output for tracking VEM computations. |
| check/debugging/r.md | Markdown report for debugging information. |
| check/debugging/vem_debug_output.md | General debug output in Markdown format. |
| check/debugging/vem_debug_output_julia.md | Julia-specific debug output. |
| check/debugging/vem_debug_output_matlab.md | MATLAB-specific debug output. |
| check/meshes/*.mat | .mat files for different mesh configurations, including L-domain, Voronoi, etc. |
docs/):| File | Description |
|———————————–|——————————————————————————|
| docs/build/index.html | HTML documentation generated with Documenter.jl. |
| docs/build/notebook.ipynb | Example Jupyter notebook. |
| docs/build/plot_solution/index.html | Plot solution documentation in HTML format. |
| docs/src/*.md | Markdown source files for the project documentation. |
| docs/make.jl | Julia script to build the documentation using Documenter.jl. |
| File | Description |
|———————————–|——————————————————————————|
| nb/vem_demo.ipynb | Example Jupyter notebook demonstrating the VEM solver. |
res/plots/):| File | Description |
|———————————–|——————————————————————————|
| res/plots/something.pdf | Example plot in PDF format. |
| res/plots/something.png | Example plot in PNG format. |
| res/plots/something.svg | Example plot in SVG format. |
The following Julia packages are required to run the Virtual Element Method solver and related scripts in this project:
Plots: For generating visualizations, such as heatmaps and wireframe plots.Colors: Provides color manipulation and schemes for plotting.Statistics: Used for basic statistical calculations.MAT: For reading and writing MATLAB .mat files, which are used for storing VEM problem data.LinearAlgebra: Provides linear algebra functions, necessary for matrix operations.SparseArrays: Supports sparse matrix representations, essential for efficiently handling large systems.To install these dependencies, run the following command in the Julia REPL:
using Pkg
Pkg.add(["Plots", "Colors", "Statistics", "MAT", "LinearAlgebra", "SparseArrays"])
### Command Line Options
| Option | Description | Required | Default Value |
|---------------------|-----------------------------------------------------------------------------|----------|---------------------------|
| `--file, -f` | Path to the `.mat` file containing the VEM problem. | Yes | N/A |
| `--plot, -p` | Plot type: choose either `"heatmap"` or `"wireframe"`. | No | N/A |
| `--ptitle, -t` | Title for the plot. | No | None |
| `--colourscheme, -c`| Color scheme for the plot. Default: `"blues"` for heatmap, `"viridis"` for wireframe. | No | `"blues"` for heatmap |
| `--azimuth, -a` | Azimuth angle for the wireframe plot. Default: `30`. | No | `30.0` |
| `--elevation, -e` | Elevation angle for the wireframe plot. Default: `30`. | No | `30.0` |
| `--solidcolour, -s` | Solid color for the wireframe plot. | No | N/A |
| `--savepath, -o` | File path to save the generated plot (e.g., `output.png`). | No | N/A |
| `--debug, -d` | Enable debug mode and output detailed information. | No | Disabled |
| `--debug_file, -D` | Path to store debug output. Default: `vem_debug_output.md`. | No | `"vem_debug_output.md"` |
### Examplse
```julia
julia vem_solver.jl --file /path/to/mesh.mat --plot heatmap --colourscheme plasma --savepath output.svg
julia vem_solver.jl --file /path/to/mesh.mat --plot wireframe --azimuth 45 --elevation 60 --solidcolour red --savepath wireframe.png
julia vem_solver.jl --file /path/to/mesh.mat --debug --debug_file debug_output.md
For detailed instructions on running the code and generating documentation, refer to the docs section.
The documentation is built using Documenter.jl. To generate the HTML documentation locally, run the following:
julia> include("docs/make.jl")