I have a relatively strong background in the theory of numerical analysis of partial differential equations (PDEs) and functional analysis, particularly applied to the numerical analysis of PDEs using the finite element method. I'm also very interested in graph theory. I was wondering if there are any theories or research areas that unify these fields and that I could explore given my background. Are there any connections between graph theory, functional analysis, and numerical analysis of PDEs that I should be aware of?
I am aware that there is another question about this, but I am interested in it being in the context of PDEs and numerical methods. Also I am looking for open questions to initiate a research project, so I am seeking topics where the intersection is significant and there are open problems.
Thank you in advance for your assistance.