# MMath&Phys Mathematics and Physics

Year of entry: 2024

## Course unit details:Approximation Theory and Finite Element Analysis

Unit code MATH46052 15 Level 4 Semester 2 Department of Mathematics No

### Overview

This course unit covers the theory of approximation and applications to the numerical solution of linear elliptic partial differential equations (PDEs) using finite element approximation methods. Such methods are universally used to solve practical problems associated with physical phenomena in complex geometries. The emphasis is on assessing the accuracy of the approximation using a priori and a posteriori error estimation techniques. Practical issues will be illustrated with MATLAB using the IFISS software toolbox.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Real Analysis A MATH20101 Pre-Requisite Compulsory
Partial Differential Equations and Vector Calculus A MATH20401 Pre-Requisite Compulsory
MATH46052 pre-requisites

Students are not permitted to take, for credit, MATH46052 in an undergraduate programme and then MATH66052 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To give an understanding of the fundamental methods and theoretical basis of approximation. To provide students with the technical tools enabling them to solve practical elliptic PDE problems using the finite element method.

### Learning outcomes

On successful completion of this course unit students will be able to:

• construct a piecewise polynomial interpolant of a functions of one variable and derive bounds on the associated approximation error;
• construct a piecewise polynomial interpolant of a function of two variables using a rectangular grid or a scattered set of sampling points and establish the degree of continuity of the resulting approximation;
• distinguish between the concepts of a weak and a classical solution of an elliptic boundary value problem and establish uniqueness of a weak solution;
• define Galerkin approximations to elliptic boundary value problems and derive a priori and a posteriori bounds for the approximation error working in standard Sobolev space norms;
• construct finite element solutions to the Poisson equation in two dimensions by using a pencil and paper and by running bespoke software;
• construct conforming finite element solutions to the biharmonic equation by using a pencil and paper and by running bespoke software.

### Syllabus

1. Piecewise polynomial interpolation in one and two dimensions.  Sobolev spaces. Weak derivatives.

Surface fitting by piecewise polynomials including the thin plate spline.

2. Finite element methods for the diffusion equation in two dimensions.

Affine mappings. Linear, bilinear, quadratic and biquadratic approximation. Finite element assembly process.

Properties of the discrete equation system. A priori error bounds: best approximation in energy.

A posteriori error bounds. Local error estimators. Self adaptive refinement strategies.

3. High-order finite element  approximation The biharmonic equation. Hermite bicubic splines. Eigenvalue approximation in one and two dimensions.

Lax-Milgram lemma.  Well-posedness. Weak formulation. Galerkin approximation.

The streamline-diffusion method.  A priori error bounds.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Endre Süli  and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003. ISBN: 978-0521007948.
• Howard Elman, David Silvester and Andy Wathen, Finite Elements and Fast Iterative Solvers, ISBN 0-19-852868-X (pbk) Oxford University Press, Oxford 2005
• Dietrich Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, ISBN 978-0-521-70518-9 (pbk) Cambridge University Press, Cambridge, third edition, 2007.

### Study hours

Scheduled activity hours
Lectures 24
Tutorials 12
Independent study hours
Independent study 114

### Teaching staff

Staff member Role
David Silvester Unit coordinator