L. C. Evans, Partial Differential Equations, Second Edition.
Graduate Studies in Mathematics, volume 19.
American Mathematical Society, Providence, RI, 2010.
Selected material from:
Chap. 5: Basic theory of Sobolev spaces.
Chap. 6: Applications of Sobolev spaces to elliptic PDEs.
This note on convergence of the finite element method in 2D.
Additionally, the appendices in the book; they cover relevant background material from real analysis, functional analysis, and measure theory. Integration on boundaries, Stokes' theorem.
See also the teaching material for autumn 2016 and autumn 2014.