Research Scholarship at the University of Newcastle (2023-2024)

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.

Poison

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 $f,g \in L_2$

Project Overview

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:

• A Julia reimplementation of the VEM as described by Oliver J. Sutton.
• Supporting MATLAB files for additional analysis and comparison.
• A paper discussing the project's objectives, methodology, and results.

Directory Structure

Here’s a complete list of the files in this repository:

Main Files:

File	Description
.gitignore	Specifies files ignored by Git.
Project.toml	Julia project dependencies.
README.md	This README file describing the repository structure.

LaTeX Files:

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.

MATLAB Files:

File	Description
MATLAB/PolygonTests.m	MATLAB file containing tests for polygon meshes in the VEM context.

Source Code (Julia):

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.

Check Directory:

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.

Documentation (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.

Jupyter Notebooks:

File	Description
nb/vem_demo.ipynb	Example Jupyter notebook demonstrating the VEM solver.

Resources (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.

cc Dependencies

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

How to Use

For detailed instructions on running the code and generating documentation, refer to the docs section.

Generating Documentation

The documentation is built using Documenter.jl. To generate the HTML documentation locally, run the following:

julia> include("docs/make.jl")