Introduction to Theory and Numerics of Partial Differential Equations

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Contents

This lecture is the first in a series of sequential lectures on the theory and numeric of partial differential equations.

Partial Differential Equations (PDEs) often serve as models for physical processes, e.g., in determining a temperature distribution, in describing vibrations of membranes, or fluid flows.

In this lecture, we will focus on elliptic differential equations. We will cover both the classical existence theory and the modern theory of solvability for such equations. Even if one has explicit solution formulas for simple problems, these can rarely be computed in practice. Therefore, it is important to compute numerically approximate solutions and to verify that they converge appropriately toward the exact solution. To this end, the lecture will present the corresponding theory of the Finite Element Method (FEM).

Lectures will be held in person, following the course lecture notes (German version here).

Programming lab exercises will be offered in parallel with the lecture.

Study and examination requirements

Exercise Groups

Only one (or at most two) of the following exercise groups will be offered. The final decision will be made on Monday, 19th October during the first lecture.

Exercise sheets

Solutions are to be submitted in pairs. The exercise sheets can be dropped off in the submission boxes in Ernst-Zermelo-Straße.

Lab exercise session

In the programming lab, students will implement the numerical methods for solving partial differential equations presented in the lecture. The goal is to develop a software package for computing approximate solutions of elliptic partial differential equations using the finite element method.

The lab requires programming experience in MATLAB (or Python). Knowledge of Numerical Mathematics I and II is strongly recommended as a prerequisite. Students planning to write a thesis in applied mathematics are strongly encouraged to participate.

The practical sessions associated with the lecture will take place every week, starting from the second week of lectures.

All sessions will be held in the CIP-Pool, Room 201 at the Rechenzentrum (Hermann-Herder-Str. 10). Please note that new exercises will be assigned every two weeks, which means the submission/deadline for the assignments is on a bi-weekly basis.

Lab exercise sheets

Solutions for the bi-weekly practical assignments must be submitted (in pairs) via e-mail. Students are required to complete at least 50% of the total available points on the lab sheets.

This guide will help you install the package DistMesh in Matlab (a Python module is also available)

Literature

1. L. C. Evans, Partial Differential Equations, AMS (1998).
2. D. Braess, Finite Elemente, Springer (2007).
3. G. Dziuk, Theorie und Numerik partieller Differentialgleichungen, De Gruyter (2010).
4. S. Bartels, Numerical Approximation of Partial Differential Equations, Springer (2016).