Vem++, a C++ library to handle and play with the virtual element method.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Scientific Computing

Background:

• The Finite Element Method (FEM) is a powerful numerical technique for solving differential equations.
• Handling complex, non-standard geometries (polytopal meshes) poses challenges for traditional FEM.
• The Virtual Element Method (VEM) offers a generalized approach to FEM for such complex meshes.

Purpose of the Study:

• To introduce the Virtual Element Method (VEM) for a 2D Laplacian problem.
• To highlight implementation differences and challenges of VEM compared to FEM.
• To present Vem++, a C++ library designed to facilitate VEM implementation and application.

Main Methods:

• Introduction to VEM for a 2D Laplacian problem.
• Comparative analysis of VEM and FEM implementation strategies.
• Development and description of the Vem++ C++ library.

Main Results:

• Identification of key implementation issues associated with the VEM framework.
• Demonstration of Vem++ as a solution to address these VEM challenges.
• Vem++ provides a flexible platform for applying VEM to diverse PDEs.

Conclusions:

• VEM offers a viable extension to FEM for complex meshing scenarios.
• The Vem++ library effectively tackles VEM implementation hurdles.
• Vem++ enables "plug-and-play" integration of new features for advanced VEM applications.