You should consider the following expenses:
Accommodation: see arriving, step 1
Health insurance: public health insurance companies in Germany charge a fixed percentage of your gross income.
Food: you can eat relatively cheaply in the university’s canteen. There are also several cafés on campus offering a range of different low priced dishes.
Computers and books: the MPI-INF and MPI-SWS offer well-equipped libraries and free access to computers so that you will be able to keep these costs down.
Public transport: If you register as a PhD student at Saarland University or TU Kaiserslautern, you receive a student id-card. This includes the “Semesterticket” that allows unlimited travel on nearly all kinds of public transport in Saarland or parts of Rhineland-Palatinate.
Rundfunkbeitrag: the Rundfunkbeitrag is a fee collected by the the state-run organization “ARD ZDF Deutschlandradio Beitragsservice” for use of radio and television under public law. Everybody who owns a radio and/or TV set or any other digital devices like computers with internet connection and smartphones must pay these fees.
Consider these just basic costs and only examples. Do not calculate too tightly, you will need extra money for everyday life (e.g. telephone, clothing, going out, etc.).