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Approximation Theory and Finite Element Analysis

Unit code: MATH46052
Credit Rating: 15
Unit level: Level 4
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

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.


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.


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.

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;
  • explain the difficulties that arise when constructing finite element approximations to convection-diffusion equations and establish a priori error bounds that show good approximation is possible if outflow layers in the solution are resolved by the mesh.


Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Mid-semester coursework: 20%
  • End of semester examination: three hours weighting 80%


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. Finite element methods for the convection-diffusion equation in one and two dimensions.

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

The streamline-diffusion method.  A priori error bounds.

Recommended reading

  • 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.

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.

Study hours

  • Lectures - 30 hours
  • Tutorials - 6 hours
  • Independent study hours - 114 hours

Teaching staff

David Silvester - Unit coordinator

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