xyuan / vem3d

Repository that hosts the implementation of the Virtual Element Method in 3 dimensions. A theorerical analysis of the method is also available.

VEM3D

VEM3D is a project for the Master courses of Advanced Programming for Scientific Computing by Prof. Formaggia and of Advanced Numerical Analysis of Partial Differential Equations by Prof. Perotto, at the Politecnico di Milano.

This is part of the Laurea Magistrale (equivalent of a Master) in Mathematical Enginnering, with a specialization in Scientific Computing.

This project has been done under the supervision of Prof. Formaggia, Prof. Perotto, Prof. Verani and Prof. Antonietti.

Team

• Stefano Savarè
• Ondine Chanon

Goals

• Implementation in C++ of the linear Virtual Element Method on Laplace equation with Dirichlet boundary condition, in 2D and in 3D, using all the power of the C++ programming so that the code could be developped further and generalized. This is done in the VEM/ folder. A Doxygen documentation is also furnished there.
• Theoretical explanation of the method, together with convergence results. This is done in the Report/ folder, the pdf format of the report is called main.pdf. In this same report is also explained our implementation of VEM.
• Convergence analysis of the linear Virtual Element Method under mesh refinement. This is done in the Convergence/ folder in Matlab. This also allows us to test the implementation.

For a sake of completeness, the following has also been made:

• A visualization of the solutions, in 2D and in 3D, made in Python in the Python/ folder.
• The generation of different meshes for testing, together with a possibility to visualize them. The different meshes can be found in the folder Mesh/. Some generations have been made in Python and can be found in the Python/ folder, while some others have been made either in C++ of in Matlab and they can be directly found in their corresponding subfolder in Mesh/.

How to use our code

More detailed README are present in each folder, with all the instructions about compiling, running and using our code.