# MA949 - Applied and Numerical Analysis for Linear PDEs

Term 1: One lecture per week will be f2f teaching on Monday 15-16 in MS.01 (large lecture theater with a capacity of 300 allowing for social distancing). In addition live online lectures will take place Thursday and Friday 11-12.

The online lectures will be recorder (if you have any issue with appearing in the recording let me know), Participation is strongly encouraged so feel free to keep microphone and camera on if you wish and ask questions anytime during the lecture.

Lecturer: Andreas Dedner

Term(s): Term 1

Commitment: 30 lectures

Assessment: Oral exam (50%), Course work (50%)

Prerequisites: Familiarity with topics covered in Multivariable Calculus and other topics covered in first and second year analysis modules

Content:

Teaching for this course will consist of lectures, as well as self-study from the notes provided and from reference books. The topics which will be covered in lectures include (details will depend on the background of the participants):

• Hilbert and Banach spaces, Lebesgue integration and the $\mathrm{L}^p$ spaces.
• Sobolev spaces and their relationship to spaces of continuous and integrable functions.
• The Riesz Representation Theorem and Lax-Milgram Lemma, and their application to elliptic PDEs.
• An overview of regularity results for weak solutions of linear elliptic PDEs.
• Introduction to Galerkin approximation and abstract error estimates
• Discussion on the implementation of Finite-Element method for solving elliptic PDEs
• Interpolation estimates for Finite-Element functions and a-priori error estimates

If time permits the course could additionally cover: weak convergence results, weak solutions to parabolic PDEs, and a-posteriori estimates for finite-element methods, theory and approximation of saddle point problems.

References:

• Lawrence C Evans Partial Differential Equations AMS

• Michael Reed and Barry Simon Modern Methods of Mathematical Physics. I. Functional Analysis. Academic Press.

• Michael Reed and Barry Simon An Introduction to Partial Differential Equations Springer.

• Dietrich Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge University Press

• S Brenner and L Ridgeway Scott, The mathematical theory of finite element methods Springer Texts in Applied Mathematics Volume 15.