Martin W. LichtS.E.W. Visiting Assistant Professor
Department of Mathematics
Center for Computational Mathematics
University of California, San Diego
mlicht ATSIGN ucsd DOT edu
Department of Mathematics
The University of California, San Diego
9500 Gilman Dr, # 0112
La Jolla, CA 92093
Publications and Preprints
- Geometric Transformation of Finite Element Methods. With Michael Holst. In Preparation.
- Smoothed Projections over Compact Riemannian Manifolds. With Snorre Christiansen. In Preparation.
- Poincaré-Friedrichs Inequalities for Complexes of Distributional Differential Forms. With Snorre Christiansen. Submitted.
- Smoothed Projections and Mixed Boundary Conditions. Accepted for publication in Math. Comp. [Download] [Arxiv link]
- Smoothed Projections over Weakly Lipschitz Domains. Accepted for publication in Math. Comp. [Download] [Arxiv link]
- Complexes of Discrete Distributional Differential Forms and their Homology Theory, Found Comput Math (2017) 17: 1085-1122. [Arxiv link] [Published version]
- On the A Priori and A Posteriori Error Analysis in Finite Element Exterior Calculus. PhD thesis in mathematics, Oslo, 2017. [Download]
- Smoothed Analysis of Linear Programming. Diplom thesis in computer science, Bonn, 2013. [Download]
- Diskrete distributionelle Differentialformen und ihre Anwendungen. Diplom thesis in mathematics, Bonn, 2012.
- Domain Distribution for parallel Modeling of Root Water Uptake. Proceedings 2010, JSC Guest Student Programme on Scientific Computing, 2010. [Link to proceedings]
My research develops around structure-preserving numerical methods for partial differential equations.
This has been a very active area of research in the past years.
The basic »paradigm« of structure-preserving numerical methods
is to mimic qualitative properties of the analytical problem on a discrete level,
since qualitative properties, such as energy conservation, are often very important
for practical applications in physics and industry.
The mathematical beauty of these methods lies in the confluence
of numerical, global, and functional analysis,
of differential geometry and algebraic topology.
I focus on finite element exterior calculus, whose main idea is to construct a de Rham complex of spaces of finite element differential forms. Not only does it provide a very powerful tool in the construction and understanding of finite element methods, but gives the background for a productive exchange in pure and applied analysis. In fact, one can discover that many similar ideas have been used in finite element analysis and global analysis.
The theory of these methods is mathematically very demanding (which might explain my passion for this research area). I am convinced this mathematics is necessary for effectively mastering complex problems in computational physics. I value thorough and detailed research, and it is my ambition to keep the big picture in perspective; in fact, keeping an eye on the details is often necessary to fully comprehend mathematics in the big picture, and to discover often surprising new insights.
Discrete harmonic vector fields on an annullus, computed with a lowest-order Nedéléc method. Once with tangential boundary condition (left), then with normal boundary conditions (right).