Ora:   16:15 - 17:15

Luogo:  Aula Volterra, Normale

Speaker:  Anna Weller
                University of Cologne

Titolo:       A spectral Galerkin method for the solution of reaction-diffusion equations on metric graphs

Abstract:

We investigate a spectral solution approach for reaction-diffusion equations on graphs
interpreted as topological space (metric graphs). Of special interest is the numerical
computation of eigenfunctions of the negative second order derivative acting on each
edge. Remarkably, it is possible to give an explicit characterization of these eigenfunctions

and corresponding eigenvalues. Moreover, for equilateral graphs, we will show
how to efficiently compute an arbitrary lower part of the spectrum using a very useful

relationship to the graph Laplacian matrix of the underlying combinatorial graph.
Finally, we can use the basis of eigenfunctions in a Galerkin approach to solve various

PDEs on metric graphs, were we here focus on reaction-diffusion equations. This
problem is motivated by a recent collaboration with the Institute of Geophysics and
Meteorology of the University of Cologne on the modeling of protein distribution in
Alzheimer’s disease together with the University Hospital Cologne1. Part of my numerical

results are joint work with Prof. Dr. Mark Ainsworth (Brown University) and
with Chong-Son Droege (University of Cologne).

References

[1] M. Ainsworth, A. Weller, A spectral Galerkin method for the solution of partial differential
equations on metric graphs, in preparation.
[2] M. Ainsworth, A. Weller, A spectral Galerkin method for the solution of reaction-diffusion
equations on metric graphs, Oberwolfach Reports, Workshop Report 36, 2021.
[3] A. Kunoth, A. Weller, T. Yilmaz, A computational brainsphere model for the simulation of
Alzheimer’s Disease. Snapshots of modern mathematics from Oberwolfach, in preparation.
[4] A. Weller, G.N. Bischof, P. Schl ̈uter, N. Richter, J. Dronse, ̈O. Onur, J. Kukolja, B. Neu-
maier, A. Kunoth, Y. Shao, T. van Eimeren, A. Drzezga, Finding new communities: A
principle of neuronal network reorganization in Alzheimers disease. Brain Connectivity,
11(3):225-238, 2021. DOI: 10.1089/brain.2020.0889

This work was supported by the Excellence Initiative of the University of Cologne