While I was looking for information on the Lewy equation, I found some introductory material on PDEs:

• Lectures notes by T. W. Körner
• Lecture notes by Joshi and Wassermann
• The course page for Methods of Numerical Simulation in Fluids and Plasmas, which includes a detailed notes on PDEs.
• The course page for Computational Methods, which includes lecture notes on PDEs and other topics.
• An essay on PDE as a unified subject