The course will be based on the lecture notes, but the following two books are recommended for additional reading.

Larsson & Thomée: Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45. Springer, 2008, ISBN: 9783540887058
Renardy & Rogers: An introduction to partial differential equations. Texts in Applied Mathematics 13. Springer, 1993, ISBN: 0387979522

If you're interested in some more advanced texts for your future research regarding finite elements, (nonlinear) elliptic/parabolic equation and Sobolev spaces, then I would recommend the books below.

Finite element bibles:

Brenner & Scott: The mathematical theory of finite element methods. Texts Appl. Math. 15. Springer, 2008, ISBN: 0387941932
Ciarlet: The finite element method for elliptic problems. Studies in Mathematics and its Applications 4. North-Holland, 1978, ISBN: 0444850287
Thomée: Galerkin finite element methods for parabolic problems. Springer Ser. Comput. Math. 25, Springer, 2006, ISBN: 9783540331216

Elliptic and parabolic equations:

Evans: Partial differential equations. Grad. Stud. Math. 19. American Mathematical Society, 2010, ISBN: 9780821849743
Roubíček: Nonlinear partial differential equations with applications. Internat. Ser. Numer. Math. 153. Birkhäuser, 2013, ISBN: 9783034805124

More advanced texts on Sobolev spaces:

Adams & Fournier: Sobolev spaces. Pure Appl. Math. 140. Elsevier, 2003, ISBN: 0120441438
Kufner, John, Fučík: Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. Noordhoff, 1977, ISBN: 9028600159