# 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
40

### Timetable

Mondays
14:05 - 14:55 (UK)

## Description

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

• 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 V: Neural networks and deep learning for PDEs

• [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 Fluidâ€“Structure Interaction, PhD thesis, Link
• [Nochetto, 2010] Why Adaptive Finite Element Methods Outperform Classical Ones, Proceedings of the International Congress of Mathematicians, Link
Assessment:
• Set of problems similar to examples in Lectures.

## Lecturer

• ### Professor Kristoffer van der Zee

University
University of Nottingham

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

• See MAGIC098 Description for Recommended Readings (, book)

## Assessment

The assessment for this course will be released on Monday 9th January 2023 at 00:00 and is due in before 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 hoursâ€™ work to attain the pass mark, which is 50%.

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