I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and $|\cdot|_{BV}$ is the standard BV semi-norm.
I am really new for this area. (before I am only working on the theoretical level of mathematical analysis, never study numerical method before...). So, could you guys recommend me some notes or book to help me to get start on finite difference method, particular for the problem I mentioned? Also, if it possible, could you give me some reference which contains the actual code, maybe matlab code, for the problem similar to what I mentioned above?
Thank you!