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
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:
- Hilbert and Banach spaces.
- Lebesgue integration and the 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.
- Weak solutions to mixed problems, e.g., to Stokes equations
- Introduction to Galerkin approximation and abstract error estimates
- Discussion on tjhe implementation Finite-Element spaces
- Interpolation estimates for Finite-Element functions and a-priori error estimates
If time permits, weak convergence methods, an introduction to the Calculus of Variations, weak solutions to parabolic PDEs, and a-posteriori estimates for finite-element methods may additionally be covered.
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.