I am unable to solve the 3D Poisson's equation of Triple Gate (TG) FinFET device. I had gone through IEEE Trans. Electron Devices paper but in most of the papers, there were only boundary conditions and final equations.
You can try to find some answers to your questions in IMEC-Academy. Try to visit: https://imec.csod.com/LMS/LoDetails/DetailsLo.aspx?loid=3e1e5d1e-e14b-435a-9086-72189a9a7531#t=1. It is a presentation of FinFET transistor in Sentaurus TCAD. There is a chance that you will find answers to your questions.