# Current announcements

- Lecture notes updated, Section 5.5 on implementation issues is now online.
- Questions for exams for Chapter 5 added.
- Lecture notes updated (9th February).
- The Tutorial of Monday, Jan 30 is again rescheduled to Monday, Feb 6 because of sickness.
- The registration for the exam is now open. Please make sure to register to examination number 340 (there is yet another open exam with the same title but less credit points). The deadline for registration is February 28.
- Questions for exams for Chapters 1-4 added.
- Lecture notes updated. First part of Chapter 5 is now online.
- Slides added here
- Solution of problem sheet 6 revised
- Show old announcements

# People

- Prof. Dr. Marlis Hochbruck (lectures)
- M.Sc. Andreas Sturm (problem classes)

# Weekly hours

2h lecture + 2h problem class (6 credit points)

# Contents and Prerequisites

Maxwell equations are a set of vector valued partial differential equations that are fundamental for the propagation of electromagnetic waves in media. In this lecture we start to derive Maxwell equations in integral- and differential form, discuss examples of material laws, boundary conditions, and study the well-posedness in suitable function spaces. For the numerical solution of Maxwell equations, we employ finite element methods for the spatial discretization. Our emphasis is on discontinuous Galerkin methods. Favorable methods for time discretization are splitting methods, (locally) implicit schemes, and exponential integrators. We construct and analyse these methods and discuss their efficient implementation.

The course is meant for advanced Master students who are familiar with the basics of finite element methods and numerical methods for differential equations. Some knowledge on functional analysis is also helpful.

# Schedule

## Lectures

Monday, | 15:45-17:15 in SR 3.061, building 20.30 |

Wednesday, | 9:45-11:15 in SR 3.061, building 20.30 |

Thursday, | 11:30-13:00 in SR 3.069, building 20.30 (alternative date) |

## Concrete dates

Please note that the dates for the lectures and problem classes may vary from week to week. The dates for the next weeks are listed below. If changes to already announced dates are required they will be highlighted by color.

cw 42: | Wednesday | 19.10. | (lecture) | and | Thursday | 20.10. | (lecture) |

cw 43: | Monday | 24.10. | (tutorial) | and | Wednesday | 26.10. | (lecture) |

cw 44: | Monday | 31.10. | (lecture) | and | Wednesday | 02.11. | (lecture) |

cw 45: | Monday | 07.11. | (tutorial) | and | Wednesday | 09.11. | (lecture) |

cw 46: | Monday | 14.11. | (lecture) | and | Wednesday | 16.11. | (lecture) |

cw 47: | Monday | 21.11. | (tutorial) | and | Wednesday | 23.11. | (lecture) |

cw 48: | Monday | 28.11. | (lecture) | and | Wednesday | 30.11. | (lecture) |

cw 49: | Monday | 05.12. | (lecture) | and | Wednesday | 07.12. | (tutorial) |

cw 50: | Wednesday | 14.12. | (lecture) | and | Thursday | 15.12. | (lecture) |

cw 51: | Monday | 19.12. | (tutorial) | and | Wednesday | 21.12. | (lecture) |

cw 02: | Monday | 09.01. | (lecture) | and | Wednesday | 11.01. | (lecture) |

cw 03: | Monday | 16.01. | (tutorial) | and | Wednesday | 18.01. | (lecture) |

cw 04: | Monday | 23.01. | (lecture) | and | Wednesday | 25.01. | (lecture) |

cw 05: | Wednesday | 01.02. | (lecture) | ||||

cw 06: | Monday | 06.02. | (tutorial) | ||||

Wednesday | 08.02. | (lecture) | and | Thursday | 09.02. | (lecture) |

# Exams

The format of the exams will be the following:

- Until the end of the semester, we will provide you with a list of possible questions for each chapter of the lecture.
- You randomly draw three questions from this list, each from another chapter. One question can be redrawn from the same chapter with the possibility to answer the original question.
- Then you are given 20 minutes for preparation (without any aid). Any notes that you prepare during this time can be used in the oral exam.
- The actual oral exam will last additional 20 minutes during which you have to answer the questions. This leaves approximatly 7 minutes for each question. If the answer is too short we expect you to present further details of the topic. In order to assure that you understand all aspects of the topic in question, you can always be asked further questions.
- The final grade will be the mean of the grades (1-6) from the three answered questions.

The list of questions can be found here: Questions

# Lecture notes

The lecture notes provided here are a draft version, since they are written when the lecture progresses. This includes corrections shortly after the corresponding topic was discussed.

We are grateful for any suggested corrections and improvements.

# Problem sheets

sheet 1, sheet 2, sheet 3, sheet 4, sheet 5, sheet 6, sheet 7

Sketch of solutionssolution 1, solution 2, solution 3, solution 4, solution 5, solution 6, solution 7

# Literature

- M. Hochbruck, lecture notes on "Finite element methods and Numerical methods for time dependent partial differential equations", KIT, 2015–2016
- M. Hochbruck, lecture notes on "Numerik I, Numerik II und Numerische Methoden für Differentialgleichungen", KIT, 2013–2016
- M. Hochbruck, A. Sturm, Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell's Equations, SIAM J. Numer. Anal., Vol 54, 2016
- S. Brenner, R. Scott, The Mathematical Theory of Finite Element Methods, Springer Texts in Appl. Mathematics, Vol 15, Springer-Verlag, 3rd ed., 2008
- D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer, 2012
- A. Kirsch, A. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations , Springer, 2015
- P. Monk, Finite element methods for Maxwell's equations, Oxford University Press, 2003
- D. Braess, Finite Elements, Cambridge University Press, 3rd ed., 2007