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 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 $\mathrm{L}^p$ 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.
• 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.

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.