MATH66041- 2010/2011

 

Specification

Aims

This course develops the rigorous study of PDEs using tools from analysis and numerical analysis.

Brief Description of the unit

We study the well posedness of the classical PDEs by semigroup and weak approximation and rigorously develop numerical approximation by the Galerkin and finite difference method. The module is theoretical and has the flavour of a pure module; proofs are given.

Learning Outcomes

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

• Precisely formulate the notion of solution for several important PDEs
• Prove rigorously existence and uniqueness of solution
• Develop the Galerkin method for numerical approximation
• Understand the concept of piecewise polynomial approximation in one and two dimensions

Future topics requiring this course unit

None

Syllabus

1. Introduction. Review of PDEs (elliptic, hyperbolic, parabolic). Finite difference method and convergence (by maximum principle) [4 lecture]
2. Elliptic PDEs and weak solutions. Hilbert spaces, inner product, Cauchy-Schwarz. Definition of weak derivative and weak solution. Examples. Riesz representation theorem and Lax Milgram Lemma. Proof of existence and uniqueness for model diffusion problem. More general models. [6 lectures]
3. Galerkin method. Best approximation in the energy norm. Finite element and spectral Galerkin. Rates of convergence. Comparison with finite difference approximation. [6 lectures]
4. Parabolic PDEs. Semigroups of operators. Method of lines. Proof of convergence. [6 lectures]

Additional Reading Material

1. Finite element methods for the Poisson equation. Affine mappings. Linear, bilinear, quadratic and biquadratic approximation. Finite element assembly process. Properties of the discrete equation system.
2. A priori error bounds: best approximation in energy, H1 error bounds. H2 regularity and singular problems.

Textbooks

• Endre Suli and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003.
• J. Robinson, Infinite Dimensional Dynamical Systems, Cambridge University Press, 2001.
• Howard Elman, David Silvester and Andy Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, Oxford 2005, ISBN 0-19-852868-X (pbk).

Teaching and learning methods

Two lectures and one examples class each week. To cover the additional reading material students should expect to do at least six hours private study each week.

Assessment

One test worth 10%

One homework assignment worth 35%

Two hour end of semester examination; Weighting within unit 55%

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Arrangements