Iterative Solvers for Partial Differential Equations (Iterative Löser für partielle Differentialgleichungen)

Summer Semester 2025

Lecturer: Christos Pervolianakis
Office: Ernst-Abbe-Platz 2, Room 3533
Email: christos.pervolianakis (AT) uni-jena.de
Dates: Tuesday and Thursday 12-14Uhr
Office hours: Tuesday and Thursday 15-17Uhr.

Description

Many physical and engineering problems are modeled using partial differential equations (PDEs). However, their solutions often cannot be expressed in closed form, requiring numerical approximation. Discretization methods such as finite differences and finite elements transform PDEs into large-scale linear systems that must be solved efficiently. This course delves into iterative solvers, essential for efficiently handling these systems, particularly for sparse and structured matrices. We will explore fundamental iterative methods such as Jacobi, Gauss-Seidel and multigrid methods, as well as minimization techniques like Krylov subspace methods. Additionally, we will examine their convergence properties and preconditioning techniques to accelerate computations.

Recommended knowledge

• Basic lectures in analysis and linear algebra (We will also discuss for functional analysis aspects during the lecture)
• Basic knowledge in a programming language (e.g. Python, Matlab, Julia ...)

Course Evaluation

• In this class, there will also be an optional project that includes both theoretical and numerical parts, as well as a short presentation of about 30 minutes (slides may be used) and a final report. The project will be assigned no later than the mid May. If you are interested, please send me an email to discuss it.
• If \(\textsf{F}\) represents the final oral exam grade and \(\textsf{P}\) represents the optional project grade, the course grade will be calculated using the following formula:
  $$ \textsf{course grade} = \begin{cases} \textsf{F}, & \text{if the optional project is not chosen} \\ \max\{\textsf{F}, 0.6\textsf{F} + 0.4\textsf{P}\}, & \text{if the project is completed} \end{cases}$$
• This means that completing the project will only positively impact your grade!
• Oral exams will be scheduled by individual appointment.

Lecture Notes:

Theoretical Exercises

• Exercise Set #1 .pdf file. The solutions (.pdf file (after the class)) will be presented in Thursday 24.04.2025 and Thuesday 06.05.2025.

Programming Exercises

• To be announced

Announcements

01.04.2025   I would like to inform you that the lectures for this course will begin on April 15, 2025, as I will be attending a conference next week. The two lectures scheduled for next week will be rescheduled in the near future.

15.04.2025   During the lectures on 24.04, 06.05, 20.05, 03.06, 17.06, and 01.07, we will solve exercises.