Numerical Methods for Metric Graphs

This book discusses the fundamentals of the numerics of parabolic partial differential equations posed on network structures interpreted as metric spaces. These so-called metric graphs frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices. The two central methods covered here are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. The solution of the latter eigenvalue problems, i.e., the computation of quantum graph spectra, is therefore an important aspect of the method, and is treated in depth. Further, a real-world application of metric graphs to the modeling of the human connectome (brain network) is included as a major motivation for the investigated problems. Aimed at researchers and graduate students with a practical interest in diffusion-type and eigenvalue problems on metric graphs, the book is largely self-contained; it provides the relevant background on metric (and quantum) graphs as well as the discussed numerical methods. Numerous detailed numerical examples are given, supplemented by the publicly available Julia package MeGraPDE.jl.

Provides an in-depth discussion of numerics for differential equations on metric graphs Discusses a finite element and spectral discretization with ready to use code and detailed numerical examples Presents an innovative method for solving quantum graph eigenvalue problems

Autorentext

Anna Weller completed her PhD in applied mathematics at the University of Cologne. In her research, she focused on diffusion problems on network-like structures, as well as their numerical solution and modeling in the human brain. Currently, she is a postdoctoral researcher at the Fraunhofer Institute for Algorithms and Scientific Computing.

Inhalt

Chapter 1. Introduction.- Chapter 2. Background: Graphs and Differential Equations.- Chapter 3. Finite Element Method.- Chapter 4. Spectral Solution Method.- Chapter 5. Computation of Quantum Graph Spectra.- Chapter 6. Numerical Results.- Chapter 7. Comparison of Finite Element and Spectral Galerkin Method.- Chapter 8. Application to the Simulation of Tau Propagation in Alzheimer's Disease.- Chapter 9. Conclusion.