MAGIC098: Adaptive Finite Element Methods

Course details

A specialist MAGIC course

Semester

Autumn 2022
Monday, October 3rd to Friday, December 9th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Mondays
14:05 - 14:55

Course forum

Visit the MAGIC098 forum

Description

Adaptive Finite Element Methods is an advanced course in numerical analysis. 
The first aim is to provide an introduction to the finite element method (FEM) itself, which is one of the most powerful methods for approximating solutions to partial differential equations (PDEs). 
The second aim is to provide an introduction to adaptive finite element methods, which are techniques that generate optimal meshes, and thereby provide the most efficient methodology to approximate solutions to PDEs.
Learning objectives:
* Discretise PDEs with FEM
* Analyse the stability and convergence of FEM
* Approximate functions using adaptive mesh refinement
* Estimate errors of approximations to PDEs
* Analyse convergence of adaptive methods
Within this course, there is a mix of methods, theory and algorithms. 

Prerequisites

Students are expected to have had an introductory course in numerical analysis. They should also have basic knowledge of partial differential equations (PDEs), and some programming experience.
(It is not assumed students are familiar with the finite element method itself.)

Syllabus

(subject to change)
  • Intro FEM I: Weak formulations of PDEs in 1-D
  • Intro FEM II: FEM discretization in 1-D
  • Intro FEM III: Well-posedness of weak formulations of PDEs
  • Intro FEM IV: FEM discretization in 2-D
  • Intro FEM V: Convergence of FEM for PDEs
  • Adaptive FEM I: Computable error estimates for FEM
  • Adaptive FEM II: Goal-oriented error estimates for FEM
  • Adaptive FEM III: Convergence of adaptive FEM for PDEs
  • Adaptive FEM IV: Optimality of adaptive FEM for PDEs
  • Adaptive FEM V: Neural networks and deep learning for PDEs

Recommended reading:
  • [Larson, Bengzon, 2013] The Finite Element Method: Theory, Implementation, and Applications, Springer Book, Link to preprint 
  • [Johnson, 1987] Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press Book, Link
  • [Ern, Guermond, 2021 b] Finite Elements II: Galerkin Approximation Elliptic and Mixed PDEs, Springer Book, Link to preprint
  • [Ainsworth, Oden, 2000] A Posterori Error Estimation in Finite Element Analysis, Wiley Interscience Book, Link
  • [van der Zee, 2004] Goal-Adaptive Discretization of Fluid–Structure Interaction, PhD thesis, Link
  • [Nochetto, 2010] Why Adaptive Finite Element Methods
    Outperform Classical Ones,
    Proceedings of the International Congress of Mathematicians, Link
  • ...more to be added...
Assessment:
  • Set of problems similar to those in Problem Sheets

Lecturer

  • Dr Kris van der Zee

    Dr Kris van der Zee

    University
    University of Nottingham

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 9th January 2023 and is due in by Sunday 22nd January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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