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
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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 end of April. If you are interested, please send me an email to discuss it.
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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:
$$ \textsf{course grade} = \begin{cases}
\textsf{F}, & \text{if the optional project is not chosen} \\
\max\{\textsf{F}, 0.6\textsf{F} + 0.4\textsf{P}\}, & \text{if the project is completed}
\end{cases}$$
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This means that completing the project will only positively impact your grade!
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Oral exams will be scheduled by individual appointment.
Lecture Notes:
The lecture notes will be uploaded here (as a .pdf file) after each class.
To access them, you must log in to Moodle using your university account.
Programming Exercises
Here you will find the some numerical exercises that are optional, but interesting.
Announcements
18.02.2025 The course lecture starts on 07.04.2025. The exersice classes will be every second Wednesday, i.e., on 16.04, 30.04, 14.05, 28.05, 11.06, 25.06