(Created 2011-09-01.)
 NUMERICAL ANALYSIS FOR ELLIPTIC AND PARABOLIC DIFFERENTIAL EQUATIONS FMNN20
Credits: 7,5. Grading scale: TH. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course will be given in English on demand. Optional for: F5, F5bs, Pi4, Pi4bs. Course coordinator: Eskil Hansen,, eskil@maths.lth.se, and Director of studies Anders Holst, Anders.Holst@math.lth.se, Numerical Analysis. Recommended prerequisits: FMNN10 Numerical Methods for Differential Equations, and started FMA260 Functional Analysis and Harmonic Analysis. Assessment: Take home exam followed by oral exam.

Aim
New and more powerful computational techiques are continuously being developed. Engineers working with computations must be able to learn, and evaluate, new algorithms.

The purpose of the course is to provide a thorough mathematical analysis of differential equations, focusing on elliptic and parabolic problems. In the basic courses in numerical analysis the emphasis is on construction och implementation of approximation methods. This course course aims to give the students an understanding of the more theoretical aspects of the subject.

By using concepts and methods from functional analysis and from the rich theory about linear partial differential equations, we will discuss existence, stability and convergence for a number of common numerical methods.

The approach to interpret both the differential equation and its numerical approximation within one and the same functional analytic framework gives a basic understanding of how numeric methods may be derived, and of how their performance is affected by the character of the original problem.

Knowledge and understanding
For a passing grade the student must

- have an understanding for how functional analytic concepts are used to develop and analyse numerical algorithms for partial differential equations.

- have developed a deeper knowledge about the interaction between type of differential equation and choice of numeric algorithm.

- have developed a good understanding for concepts such as stability and convergence.

Skills and abilities
For a passing grade the student must

- be able to derive simple error estimates.

- be able to identify important classes of partial differential equations, and be able to exploit this to efficiently discretize given equations.

- be able to give examples of important applications in which algorithms discussed in the course are of significance.

Judgement and approach
For a passing grade the student must

in simple cases, be able to balance complexity of the model against computability to obtain good accuracy.

Contents
Error estimates, convergence and stability. Existence and regularity of solutions of ordinary, elliptic and parabolic differential equations. Analysis of finite differences and finite element method. Analysis of time-stepping methods, such as implicit Runge-Kutta methods. The interaction between the discretizations in space and time. Applications of partial differential equations, such as heat conduction and diffusion-reaction processes.

Literature
Larsson, S & Thomée, V, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45, 2nd ed. Springer 2008. ISBN 978-3540887058.