Lawrence C. Evans: Partial Differential Equations, 1998. American Mathematical Society. ISBN: 0-8218-0772-2 (alk. paper). Or a later edition. Selected material from Chapter 5 and 6. Appendices as needed. The appendices in Evans cover well the material from real analysis, functional analysis and measure theory which is needed for the course. These appendices are A.1. A.2, A.3, A.4, A.5, A.6, B.2 c e f h i, C.2, C.3 (just Theorem 4), C.4, C.7, D.1, D.2, D.3 a (just the part on linear functional analysis), D.3 b (just Riesz representasjon's theorem), E.1, E.2. E.3.
Kendall Atkinson and Weimin Han: Theoretical numerical analysis, 2009. Springer. ISBN: 978-1-4419-0457-7. Selected material from Chapter 7, 8, 9 and 10. Available online.
Key notions to be covered:
Sobolev spaces: weak derivatives, density of smooth functions, traces on the boundary, extensions outside the domain. Gagliardo-Nirenberg-Sobolev injections. Rellich-Kondrachov compactness.
Elliptic equations: Lax-Milgram, Fredholm alternative, eigenvalues.
Numerical methods: Galerkin methods, finite elements.