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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.