Martin Licht

Bernoulli Instructor at EPFL, Switzerland

  • Martin Licht
    EPFL SB MATH MATH-GE
    Station 8
    CH-1015
    Switzerland
Martin W. Licht

I am a postdoctoral researcher at the Department of Mathematics at EPFL, Switzerland. My research focuses finite element methods for partial differential equations in electromagnetism, elasticity, and relativity.

After my PhD at the University of Oslo, I was a visiting assistant professor at UCSD. I have been visitor at the University of Minnesota, Cambridge University, and Brown University.

In my free time, I paint with acrylics and learn Mandarin.

Education

01/2014 - 06/2017 PhD in Mathematics, University of Oslo.
Thesis: On the A Priori and A Posteriori Error Analysis in Finite Element Exterior Calculus. [Download] [Thesis Abstract]
10/2006 - 12/2013 Diplom (master's degree) in Computer Science, University of Bonn.
Diplom thesis: Smoothed Analysis of Linear Programming. [Download]
Final grade: "sehr gut".
10/2006 - 08/2012 Diplom (master's degree) in Mathematics, University of Bonn.
Diplom thesis: Discrete distributional differential forms and their applications.
Final grade: "sehr gut".
07/2018 - 08/2018 Micro-MBA, Rady School of Management, UCSD.

Positions & Activities

Since 09/2021 Bernoulli Instructor, EPFL, Switzerland.
01/2021 - 06/2021 Postdoctoral Scholar, UC San Diego, USA.
09/2020 - 12/2020 Postdoctoral Scholar, ICERM, Brown University, Providence, USA
Program Advances in Computational Relativity
07/2019 - 12/2019 Visitor at Isaac Newton Institute, University of Cambridge, UK.
Program Geometry, compatibility and structure preservation in computational differential equations
07/2017 - 06/2020 S.E.W Visiting Assistant Professor, UC San Diego, USA.
09/2015 - 05/2016 Visiting Research Scholar, University of Minnesota.
06/2010 - 09/2010 Visiting Scholar, Jülich Supercomputing Centre.
Domain Distribution for parallel Modeling of Root Water Uptake. Proceedings 2010, JSC Guest Student Programme on Scientific Computing, 2010. [Link]

Publications

Preprints

Research Interests

Why do we need edge elements?

Edge elements are necessary for physically correct finite element approximations in numerical electromagnetism. This inevitable leads to the use of mixed finite element methods. By contrast, finite element methods that naively use Lagrange elements for the coordinates and minimize energy functionals stably converge to incorrect solutions. This is catastrophic because the inconsistency may not be apparent to every user of finite element methods.

ExampleWhitney

The finite element solution using edge elements correctly resolves the corner singularity of the problem. The plot depicts the vector field variable and the magnitude of the solution. As the mesh is refined, the finite element approximation converges to the physically correct solution.

ExampleLagrange

A boilerplate finite element method, simply using Lagrange elements for each coordinate, converges to an incorrect solution as the mesh is refined. While this vector field may look permissible to the untrained eye, it is qualitatively different from the correct solution.


Discrete harmonic vector fields

MixedBC

Discrete harmonic vector fields on a square subject to mixed boundary conditions: we impose tangential boundary conditions along the middle parts of the four boundary faces and normal boundary conditions near the corners.

Even though the domain is simply connected, the space of harmonic vector fields is non-trivial. Its dimension is the first relative Betti number of the domain, which equals 3 in this example.

The low elliptic regularity of the vector Laplace equation in the presence of mixed boundary conditions is apparent.


Anulus Anulus

Discrete harmonic vector fields on an anulus, computed with a lowest-order Nedéléc method. Once with tangential boundary condition (left), then with normal boundary conditions (right). The space of harmonic vector fields reflects the non-trivial topology of the domain.


Complexity of adaptive mesh refinement

RoundCoarse RoundFine RoundNVB

Triangulations of an approximate sphere. My research has led to the first completely combinatorial amortized complexity estimate for repeated newest vertex bisection. Newest vertex bisection is a key component in the mesh refinement algorithm of adaptive finite element methods. Previous complexity estimates involved geometric quantities that lead to suboptimal estimates when the initial mesh is highly non-uniform.

Structure-Preserving Numerical Methods for Partial Differential Equations

Bernoulli Center, Lausanne, Switzerland, July 3-7, 2023

https://suprenumpde2023.epfl.ch/

I have had the distinguished pleasure to co-organize this international workshop with over 40 participants and with five highly renowned keynote speakers.
Martin W. Licht

Selected Presentations

Undergraduate Supervision


Teaching

«Great! Easy to understand despite French accent and explains examples very well!» - quote from one of my teaching reviews...

Miscellaneous recommendations

Code