Computational PDEs

Studying

Courses

Members of the working group regularly teach courses in numerical mathematics and the numerics of partial differential equations. Current offerings are listed in UniVZ.

Teaching history (Christoph Lehrenfeld) ▸
  • WS 24/25 (University of Göttingen) — see UniVZ
  • SS 2024 (University of Göttingen) — see UniVZ
  • WS 23/24 (University of Göttingen) — see UniVZ
  • SS 2023 (University of Göttingen) — see UniVZ
  • WS 22/23 (University of Göttingen) — see UniVZ
  • WS 21/22 (University of Göttingen)
    • Mathematik für Informatik Anfänger I
    • Seminar on Good Scientific Practice
    • (Ober)Seminar on Numerical Methods for PDEs
  • SS 2021 (University of Göttingen)
    • Numerical methods for PDEs IV
    • Adv. Seminar on Numerical Methods for PDEs
    • (Ober)Seminar on Numerical Methods for PDEs
  • WS 20/21 (University of Göttingen)
    • Numerical methods for PDEs III
    • (Ober)Seminar on Numerical Methods for PDEs
  • SS 20 (University of Göttingen)
    • Numerical methods for PDEs II
    • Seminar on Numerical Methods in Fluid Dynamics
    • (Ober)Seminar on Numerical Methods for PDEs
  • WS 19/20 (University of Göttingen)
    • Numerical methods for PDEs I
    • (Ober)Seminar on Numerical Methods for PDEs
  • SS 19 (University of Göttingen)
    • Numerical methods II
    • (Ober)Seminar on Numerical Methods for PDEs
  • WS 18/19 (University of Göttingen)
    • Seminar on Scientific computing with partial differential equations
    • (Ober)Seminar on Numerical Methods for PDEs (joint with Prof. Lube)
  • SS 18 (University of Göttingen)
    • Scientific computing with partial differential equations
    • (Ober)Seminar on Numerical Methods for PDEs (joint with Prof. Lube)
  • WS 17/18 (University of Göttingen)
    • Seminar on Numerics of partial differential equations
    • (Ober)Seminar on Numerical Methods for PDEs (joint with Prof. Lube)
  • SS 17 (University of Göttingen)
    • Numerics of partial differential equations
    • (Ober)Seminar on Numerical Methods for PDEs (joint with Prof. Lube)
  • WS 16/17 (University of Göttingen)
    • Advanced Finite Elements
    • (Ober)Seminar on Numerical Methods for PDEs (joint with Prof. Lube)
  • SS 16 (WWU Münster)
    • Numerical methods for PDEs II
    • Seminar on Numerical Methods in Biological and Medical Sciences
  • WS 15/16 (WWU Münster)
    • Numerical methods for PDEs I
    • Practicals to Numerical methods for PDEs I
    • Introduction to programming with Python
  • SS 15 (TU Wien) — Extended Finite Element Methods
  • WS 14/15 (RWTH Aachen) — Seminar on Numerical Analysis
  • SS 13 (RWTH Aachen) — Numerical Mathematics for Mech. Engineers
  • WS 10/11 (RWTH Aachen) — C++-Course
  • SS 09 (RWTH Aachen) — Advanced Finite Element Methods
  • SS 08 (RWTH Aachen) — Mathematische Grundlagen II (CES)

Student projects

Topics for Bachelor's and Master's theses as well as scientific-computing practicals are available throughout the year and upon request. Projects typically involve a mix of mathematical analysis, algorithm design, and implementation in NGSolve / ngsxfem / ngstrefftz, on our research themes: structure-preserving flows, unfitted finite element methods, wave propagation and Trefftz methods. Just get in touch with Christoph Lehrenfeld to discuss current availability and your interests.

This list is being updated — the topics below mix concrete current openings with past ideas, so please contact us directly for the latest.

Current open topics

Scientific computing practical Draft
Mean curvature computation in unfitted FEM

Carry the variational (distributional) mean-curvature lifting of Gopalakrishnan, Neunteufel et al. from the fitted to the unfitted (level-set) setting, with ghost-penalty stabilization. Implemented in NGSolve / ngsxfem.
Ba / Ma thesis or practical Draft
Moving Stokes with pointwise divergence-free H(div)-FEM

Fuse the Eulerian unfitted moving-domain method with an H(div)-conforming Stokes discretization to obtain exactly divergence-free and pressure-robust solutions.
Ba / Ma thesis Draft
Axisymmetric Stokes with an H(div) discretization

Construct H(div)-conforming elements adapted to cylindrical coordinates, achieving pointwise divergence-freeness without the reconstruction post-processing of the group's prior work.
Master's thesis Draft
Analysis of the continuous-in-time space-time unfitted FEM

Prove stability and optimal-order error estimates for the continuous Petrov–Galerkin (cPG) variant of the isoparametric space-time unfitted FEM, extending the existing DG-in-time analysis.
Scientific computing practical Draft
Axisymmetric PDEs on moving domains

Add a rotationally symmetric setup to ngsxditto's unfitted moving-domain machinery — enabling 3D axisymmetric problems (e.g. a rising bubble) at 2D cost. Primarily an implementation & testing task.

Further thesis topics

  • Weak Trefftz DG methods (analysis, unfitted Trefftz DG, …)
  • Trefftz DG methods for Stokes and Navier-Stokes equations
  • Embedded Trefftz DG methods for linear advection equations
  • Numerical solution of a vectorial model problem of solar oscillations
  • Discretization of incompressible flows on moving surfaces
  • A combination of Trefftz DG and Trace-DG for Stokes problems on unfitted surfaces
  • Unfitted space-time discretization for Stokes and Navier-Stokes flow on moving domains
  • Trefftz DG methods (introductory)
  • Preconditioning of a Helmholtz-Beltrami operator on a sphere

Scientific computing practicals (further ideas)

Suitable also for students without a strong background in PDE numerics:

  • Reinitialization / redistancing of level-set functions with k-d-trees (Python or C++)
  • ODE-solver framework for implicit-explicit time integration schemes (Python)
  • Time-adaptivity with space-time discretizations of parabolic PDEs (Python)

Finished theses

Completed theses are listed on the publications page (Master's) and publications page (Bachelor's).

Student jobs

We occasionally look for student assistants. Open positions are announced here and via the institute; feel free to enquire directly.