As a general comment, when preparing your application I'd recommend avoiding too bold
"General purpose Finite Element library and PDE solver: I have very little idea about algorithms for PDE and finite elements. But I think I can read them and with your help understand them quickly."
It usually take two full semesters for a master's student majoring in computational mathematics
to get started with these topics, it's hard to believe you'd be able to catch up before GSoC if you're starting from scratch.
I'm happy to hear of your interest in GNU Octave. I would definitely
encourage you to develop your project ideas and apply to GSoC. Having
contributed in some way to Octave or other numerically oriented
open-source projects, though not required, would make your candidacy