Ciarlet's text is not the only book to analyze in depth the mathematical theory of finite element methods, but it is still one of the best. The mathematics in the book is mostly self-contained beyond some facts of functional analysis, but readers without an appropriate level of mathematical maturity will find it difficult. The main focus of the book is on elliptic problems, and particularly linear problems. Those readers who want to deal with eigenvalue problems, constrained problems in which ellipticity fails, problems in dynamics, or problems with strong nonlinearities will probably not find all they want in Ciarlet; however, the book has an excellent bibliography with many pointers to additional literature.
