Numerical methods for solving PDEs on graphs

Michele Benzi, Emory University

11:00 - 11:40

There is currently considerable interest in a class of models (known as Quantum Graphs) which can be described in terms of PDEs posed on large and possibly complex graphs. Such models have found applications in quantum chemistry, solid state physics, neuroscience, network flow, and other areas. Both boundary and initial value problems are of interest in applications, as well as eigenvalue problems. In this talk I will present some methods for solving simple model PDEs on graphs. Discretization of PDEs posed on graphs using (say) finite elements and implicit time stepping techniques leads to sparse systems of algebraic equations of huge size which, however, possess favorable properties for iterative solution of the reduced-order system obtained by Schur complement reduction. The use of a preconditioned conjugate gradient method leads to optimal solution complexity in the elliptic case. Some results for the problem of heat diffusion on graphs will also be discussed.

This is joint work with Mario Arioli.