# XFEM

## Contents

# Dates (SS2015)

The lecture has been held in SS 2015 at TU Wien.

Lecturer: Christoph Lehrenfeld

Poster, TISS, organisational notes (updated), lecture notes

Further information: Prof. Dr. Joachim Schöberl

First meeting: Tue, 3. Mar, ~~ 15:00 ~~ 13:30 , Besprechungszimmer 3. Stock (vis a vis SR 101B)

**Lecture+Exercise: Thursday 16:00 - 18:30, Besprechungszimmer 3. Stock (vis a vis SR 101B) **

# Content

In the lecture modern numerical methods for the discretization of interface problems are presented. Partial differential equations the solution of which has (weak) discontinuities across given interfaces require special care for their numerical treatment. The concept of extended finite element methods (XFEM) introduces finite element spaces which provide ansatz functions which are discontinuous across the interface. Based on this approach the XFEM is used for different applications. We will present interface problems arising in two-phase flow applications and discuss their numerical solution. The lecture includes

- the introduction of XFEM,
- the discussion of implementational aspects
- and their numerical analysis.

The students will work on a student project. Presentation of the problem setting and their solution is part of the course.

## Student projects

### The Trace FEM for PDEs on surfaces

Project report by T.Jawecki, M. Wess

### A fictitious domain method for an osmosis problem

Project report by M. Hochsteger, C. Lackner, L. Kogler

### Two-phase Stokes with velocity/pressure XFEM

Project report by P. Lederer, C. Pfeiler, C. Wintersteiger

### XFEM discretization of a stationary Stefan problem

Project report by A. Rieder, M. Ruggeri , B. Stiftner

## Exercise material / Netgen/NGSolve tutorials

Debian packages for Netgen-6.1-dev and NGSolve-6.1-dev

### Exercise schedule / content

- Notes on how to get started with the installation of Netgen/NGSolve, html, pdf
- March, 12th - 26th, 2015: Introduction to Netgen/NGSolve:

- April, 16th-23th, 2015: Introduction to Netgen/NGSolve/NGSxfem

- Exercise material: content(tar-ball)

### Other sources for learning to deal with Netgen/NGSolve

- Concepts used in NGSolve:lecture notes 'Scientific Computing'(2004/2005)

- Slides by Christoph Lehrenfeld (Introduction to Netgen/NGSolve):
- NGSolve's pde definition (including geometry): presentation (2009)
- Programming with NGSolve:presentation (2009)

- Slides by Jay Gopalakrishnan (advanced usage of NGSolve):
- "NGSolve::Give me your element?": presentation (2015)
- "NGSolve::What's under the hood?": presentation (2015)
- "NGSolve::Come to the edge": presentation (2015)
- "NGSolve::Take the rough with the smooth": presentation (2015)

## Lecture material / literature

Organisational notes:

Lecture notes will be created during the semester.

- Lecture notes (last update: June, 4th, 2015), pdf

Further reading (on which the lecture notes are also based):

- Sven Groß, Arnold Reusken, Numerical Methods for Two-Phase incompressible flows, Springer textbook
- Christoph Lehrenfeld, On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems, Ph.D. thesis, RWTH Aachen, 2015, pdf(A4), pdf(A5)