MA949  Applied and Numerical Analysis for Linear PDEs
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 selfstudy 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 LaxMilgram 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 FiniteElement spaces
 Interpolation estimates for FiniteElement functions and apriori error estimates
If time permits, weak convergence methods, an introduction to the Calculus of Variations, weak solutions to parabolic PDEs, and aposteriori estimates for finiteelement methods may additionally be covered.
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.