The PolyDiM library

Table of Contents

• Methods
• Features
• GeDiM
• External libraries

PolyDiM (POLYtopal DIscretization Methods) is a numerical computational library designed for solving partial differential equations (PDEs) using discretization methods that operate on generic polytopal (polygonal/polyhedral) meshes.

PolyDiM is inspired by and built upon the foundational research of the Numerical Analysis Group in the Department of Mathematical Sciences "Giuseppe Luigi Lagrange" (DISMA) at the Politecnico di Torino.

Methods

The following methods have been implemented and tested:

• FEM (Finite Element Method) - A widely used numerical method for solving partial differential equations (PDEs), which discretizes a continuous domain into smaller local subdomains (finite elements).
• VEM (Virtual Element Method) - An advanced numerical method that extends FEM to general polygonal and polyhedral meshes, offering greater flexibility and high accuracy.

Features

The library's key features include:

• High-Order Approximation: supports arbitrary polynomial degrees for highly accurate solutions.
• General Mesh Support: efficiently handles both polygonal and polyhedral meshes.
• Modular and Extensible: a well-structured framework enables users to define custom problems, mesh types, and solvers.
• Applications: suitable for solving linear and nonlinear elliptic PDEs, including Poisson, Elasticity, and Navier-Stokes problems.

GeDiM

PolyDiM is fully integrated with the GeDiM [8] library (GEometry DIscretization Methods), which is developed by the same authors. This library provides the core functions and classes responsible for computing the geometric properties of the mesh elements used in the PDE solution process. Additionally, it offers solvers and a wide range of tools for mesh generation, along with an extensive collection of methods to refine and agglomerate mesh elements.

External libraries

PolyDiM currently interfaces and exploits the following libraries:

• Triangle
• TetGen
• Voro++
  
• BLAS
  
• LAPACK
  
• Eigen
  
• SuiteSparse
  
• PETSc
  
• METIS
  
• VTK

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Installation Guide