Here are some notes on finite difference schemes for PDEs. Some of these were produced rather quickly so there may be errors scattered throughout (especially with some of the calculations). The last section is incomplete/incorrect. It contains a little more than I wanted to know about Laplace transforms but unfortunately not enough of what I wanted to know about boundary conditions....
Mostly what is in here is some discussion about Poincare type estimates which come up when attempting to apply the Lax-Milgram Lemma. I've derived the one-dimensional versions of these inequalities. If nothing else, they should serve as a memory aid.