MeGraPDE

MeGraPDE stands for MetricGraphPDEs and implements numerical methods for the solution of partial differential equations (PDEs) on metric graphs.

I have developed the MeGraPDE.jl package in connection with my Ph.D thesis at the University of Cologne [W].

Among others, the package includes

construction of a variety of exemplary metric graphs and test problems
discretization via extended graphs
finite element solver for PDEs on metric graphs, e.g. in combination with a multigrid approach
computation of quantum graph eigenvalues and eigenfunctions
spectral Galerkin solver for PDEs on metric graphs, e.g. in combination with a filon-quadrature

The package relies on the methods from Graphs.jl for combinatorial graphs.

The finite element discretization via extended graphs is implemented based on the original work [AB]. The computation of equilateral quantum graph eigenvalues is based on an idea originally proposed by von Below [B]. The remaining methods and the related theory have been derived for [W] and are discussed therein.

The package is under continuous development.

[W] Anna Weller, Numerical Methods for Parabolic Partial Differential Equations on Metric Graphs, PhD thesis at the University of Cologne, 2024.

[AB] Mario Arioli, Michele Benzi, A finite element method for quantum graphs, IMA Journal of Numerical Analysis, Volume 38, Issue 3, July 2018, Pages 1119–1163.

[B] Joachim von Below, A characteristic equation associated to an eigenvalue problem on c2-networks. Linear Algebra and its Applications, 71:309–325, 1985.

Installation

The package can be added by specifying the URL to the Git repository. In your julia terminal, enter the following commands

julia> using Pkg
julia> Pkg.add(url="https://github.com/AnnaWeller/MeGraPDE.jl");

You are all set. The package can now be activated with the command

julia> using MeGraPDE