MA949-15 Applied and Numerical Analysis for Linear PDEs
Introductory description
The module will introduce students to the analysis and numerical approximation of variational solutions of linear PDEs
Module aims
Students will learn about the variational formulation of linear PDEs, how to prove existence and uniqueness of weak solutions, and how to approximate these problems using Galerkin methods.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
- 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.
- Galerkin approximations
- Introduction to Finite element methods
- Approximation theory and a-priori error analysis
Learning outcomes
By the end of the module, students should be able to:
- After completing this module, students will be able to: Recall the definition of the Sobolev spaces and their connection to other function spaces Formulate notions of weak solutions for linear elliptic PDEs. Apply Hilbert space theory to prove existence of weak solutions to linear elliptic PDEs. Use finite element methods in order to numerically solve linear elliptic PDEs Prove a-priori error estimates for finite element approximations
Indicative reading list
Elliptic Partial Differential Equations of Second Order. (2nd edition). D Gilbarg, N S Trudinger, Springer.
Sobolev Spaces (Second Edition). Robert A Adams and John J F Fournier, Academic Press.
Dietrich Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (3rd edition), Cambridge University Press (2007).
S Brenner and L Ridgeway Scott, The mathematical theory of finite element methods (3rd edition), Springer Texts in Applied Mathematics Volume 15 (2008).
Subject specific skills
- Develop a deep understanding and applicability of the variational formulation of linear PDEs,
- how to prove existence and uniqueness of weak solutions, and
- how to approximate these problems using Galerkin methods
Transferable skills
- sourcing research material
- prioritising and summarising relevant information
- absorbing and organizing information
- presentation skills (both oral and written)
Study time
| Type | Required |
|---|---|
| Lectures | 30 sessions of 1 hour (20%) |
| Private study | 120 hours (80%) |
| Total | 150 hours |
Private study description
Review lectured material.
Work on suplementary reading material.
Source, organise and prioritise material for additional reading.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group A
| Weighting | Study time | |
|---|---|---|
| Course Project | 50% | |
|
An essay (about 16 pages) on a topic discussed between the lecturer and the student. |
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| Oral Exam | 50% | |
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An oral exam involving a presentation by the student, followed by questions from the panel (2 members of the department) |
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Feedback on assessment
A copy of the essay with comments will be returned to the student.\r\nStudents will receive feedback from the course instructor after the oral exam, to cover also areas like presentation skills and use of technologies (or blackboard)
There is currently no information about the courses for which this module is core or optional.