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 covered in this module will depend on the background of the participants and will be discussed in detail in the first lecture.

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.
• 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, more general Galerkin approximation, e.g., Discontinuous Galerkin methods.

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.