# Mathematical and numerical techniques for PDE

NEWS: Deadline for handing in remaining homework problems: February 20!

This course is given at graduate level and corresponds to 10 hp. The goal is to give an overview of important numerical and mathematical concepts related to PDEs, but also to discuss in detail some techniques, such as analysis of boundary conditions in continuous and discrete settings, and how to use linearization to develop a non-linear theory. Some knowledge of function spaces and functional analysis is useful.

There will be 10 lectures, 4 sets of homework problems, and 1 research papers to read and present.

##### literature:

[1] Gustafsson, Kreiss, Oliger, Time dependent problems and difference methods, 2nd edition, Wiley 2013

[2] Kreiss, Lorenz, Initial-Boundary value problems and the Navier-Stokes Equations, SIAM 2004

[3] Evans,Partial Differential Equations, AMS

[4] Some FEM text, for example Grossmann,Roos,Stynes, Numerical Treatment of Partial Differential Equations (Ch 4.1-4.4)

For more on hyperbolic conservation laws see

[5] LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge 2002

##### teacher:

First lecture: October 8, 10.15-12 in room 6102

## outline of the course:

date&time | room | topic | reading | problem set | |
---|---|---|---|---|---|

1 | Oct 8, 10-12 | 6102 | Introduction and discussion of mathematical properties of different types of PDEs | Ch1 in [1] | problem set 1 due October 27 |

2 | Oct 17, 13-15 | 6102 | Wellposedness and stability for time-dependent PDE | Ch3&4 in [1] | |

3 | Oct 22, 10-12 | 2344 | more about wellposedness and stability | Ch3 in [2] | |

4 | Oct 29, 10-12 | 2344 | elliptic PDEs | Evans Ch6 | problem set 2 due December 1 |

5 | Nov 6, 10-12 | 2345 | more on elliptic | ||

6 | Nov 19, 10-12 | 2345 | Non-linear example | Ch4&5 in [2] | |

7 | Dec 10, 10-12 | 1111 | Non-linear example | Ch 7 in [1] | problem set 3 due January 14 |

8 | Dec 17, 10-12 | 2345 | IBVP:energy method | Ch 8&11 in [1] | |

9 | Jan 14, 10-12 | 2345 | IBVP:normal mode analysis | Ch 9&12 in [1] | problem set 4 due January 30 |

10 | Jan21, 10-12 | 2345 | error estimates | ||

Jan 30, 10-12, 13-16 | paper presentations and discussions | List of suggested papers |

Paper presentations: January 30. I expect everyone to attend all presentations.