(Created 2009-08-11.)
 ADAPTIVE METHODS FOR DIFFERENTIAL EQUATIONS FMN135

Higher education credits: 7,5. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in English on demand. Optional for: E4, F4, F4tvb, Pi4, Pi4bs. Course coordinator: Achim Schroll, Achim.Schroll@na.lu.se, Numerisk analys. Recommended prerequisits: Basic understanding of variational problems and Galerkins method. The course might be cancelled if the numer of applicants is less than 10. Assessment: Homework reports and (possibly) oral exam. Home page: http://www.maths.lth.se/na/courses/FMN135.

Aim
The aim of the course is to give an introduction to modern FE methods within a broad spectrum of applications. The focus is on goal oriented error control via duality. Goal oriented error control is a modern German-Swedish development which has become a key technology in complex technical applications

Knowledge and understanding
For a passing grade the student must

- demonstrate knowledge on numerical difficulties in adaptive FE methods

- understand aposteriori errorestimates via duality and goal oriented adaptivity

Skills and abilities
For a passing grade the student must

- independently be able to apply goal oriented adaptive FEM to the Poisson equation as well as eigenvalue problems

- be able to estimate accuracy based on local error indicators

- be able to adjust software to varying problems

Judgement and approach
For a passing grade the student must

- write a logically well structured report in suitable terminology on the construction of basic mathematical models and algorithms.

- write an algorithmically well structured report in suitable terminology on the adaptive and goal oriented numerical approximation of partial differential equations.

Contents
A model problem, FE approximation, error estimates for output functionals, goal oriented mesh adaption, higher order finite elements, practical aspects, Galerkin approximation of nonlinear problems. Applications: Eigenvalue problems, time dependent PDEs (heat equation, wave equation), applications in structural and fluid mechanics.

Literature
W. Bangerth, R. Rannacher: Adaptive Finte Element Methods for Differential Equationms, Lectures in Mathematics ETH Zurich, Birkhäuser, 2003.