Numerics IV (Advanced finite element techniques)

News

• Exercise 1 and the accompanying material can be found below.

Dates

General Information

Description

The lecture will discuss advanced aspects of finite element (FE) discretizations for partial differential equation. This incorporates theoretical aspects as well as implementation aspects. On the theoretical side we will e.g. discuss discontinuous Galerkin methods which, among other advantages, allow for highly efficient higher order methods. Another aspects will be C1 finite elements for fourth order problems. On the practical side we will discuss the impelemntation of finite elements in C++. We will cover some basics of C++, prototypical implementations, and finally introduce into the modern FE library Dune.

Target Audience

This lecture is based on the course "Numerical methods for partial differential equations (Numerik III)". It is intended to broaden the way towards a master thesis in the field of computational PDEs.

Prerequisites

Participants should have some knowledge about PDEs and their numerical approximation by finite elements as provided, e.g., by the course on "Numerical methods for partial differential equations (Numerik III)". Computer Science students interested in numerical software develpoment with C++ are also welcome.

Registration

All participants should register at the KVV, so that we know who is participating. The overall number of participants is also necessary to justify the equipment of the course. In addition, depending on your program of study, you have to register in the Campus Management.

Exercises and Criteria for a Certificate

Examination

There will be an oral exam at the end of the semester.

Criteria for the Certificate

Necessary and sufficient for a certificate are:

• exam: passing the oralexam
• active participation in the tutorial
• regular participation in the tutorial

Certificates are graded according to the result of the exam.

Exercises

• Exercise 1: Install Dune and the example programs and make yourself familiar with the examples.

  The build.sh script contains a list of commands that download/clone the necessary Dune modules and build them using the given option file. The source files of the example programs are contained in dune-fu-tutorial/src. The corresponding compiled programs can be found in the build directory under the same path. Once all modules have been build, you can recompile after possible modifications of the source files by calling make in the build directory. Four option files are provided for debug and release builds with different compilers: debug-clang.opts, debug-gcc.opts, release-clang.opts, release-gcc.opts.

Literature

• D. Braess: Finite Elemente. Springer, 3rd edition (2002)
• P. Knabner, L. Angermann: Numerik partieller Differentialgleichungen. Springer (2000)
• P. Deuflhard, M. Weiser: Numerische Mathematik 3. De Gruyter (2011)
• J. Wloka: Partielle Differentialgleichungen. Teubner (1982)
• D. Werner: Funktionalanalysis. Springer, Berlin (2000)
• H. Alt: Lineare Funktionalanalysis. Sprinter, 6th edition (2012)
• W. Rudin: Functional Analysis. McGraw-Hill, 2nd edition (1991)
• L. Evans: Partial Differential Equations. AMS, 19th volume (1998)
• F. John: Partial Differential Equations. Springer (1982)
• M. Renardy, R. C. Rogers: An introduction to partial differential equations. Springer, 2nd volume (2004)
• A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer (2002)

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