Welcome!

I am happy that you have found your way to my website. Here I want to share some of my interests and information about my work with you. In particular, I give you an outline of my research activities.

Feel free to contact me, I am always interested in productive exchange of ideas.

-- Martin

Current Position

I am currently a S. E. Warschawski Visiting Assistant Professor at the University of California, San Diego.

I received a Diplom degree in Mathematics in Bonn in 2012, with a thesis on finite element exterior calculus under supervision of Prof. Sören Bartels. One year later, I finished another Diplom degree in Computer Science with a thesis on smoothed analysis in linear programming under supervision of Prof. Heiko Röglin.

Consecutively, I was a PhD student at University of Oslo in my third year. My supervisor has been Prof. Snorre H. Christiansen. During that time I visited the group of Prof. Douglas N. Arnold at the University of Minneapolis from Fall 2015 to Spring 2016.

Research Interests

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.

Scientific Publication & Preprints

Note: I will be glad to provide drafts of submitted articles on request.

  1. Finite Element Exterior Calculus over Manifolds. with Snorre H. Christiansen. In Preparation

  2. Poincaré-Friedrichs Inequalities of Complexes of Discrete Distributional Differential Forms. with Snorre H. Christiansen. Submitted

  3. Smoothed Projections and Mixed Boundary Conditions. Submitted

  4. Smoothed Projections over Weakly Lipschitz Domains. Submitted [Arxiv]

  5. Complexes of Discrete Distributional Differential Forms and their Homology Theory. Found Comput Math (2016). doi:10.1007/s10208-016-9315-y [Arxiv]

Theses & Proceedings

  1. On the A Priori and A Posteriori Error Analysis in Finite Element Exterior Calculus.
    My PhD Thesis was supervised by Prof. Dr. Snorre Christiansen at the University of Oslo.

  2. On Discrete Distributional Differential Forms and their Applications.
    My diplom thesis in mathematics outlines finite element exterior calculus and introduces the notion of discrete distributional differential form. This thesis served as the basis for a subsequent publication. It was supervised by Prof. Dr. Sören Bartels at the University of Bonn.

  3. Smoothed Analysis of the Simplex Method.
    My diplom thesis in computer science was written under supervision of Prof. Dr. Heiko Röglin at the University of Bonn. The simplex method is the standard algorithm in linear optimization, but its practical success is not accurately by the worst-case analysis in theoretical computer science. The smoothed analysis by Spielman and Teng facilitates a probabilistic error analysis which reflects the observed practical feasibility.

  4. Domain Distribution for parallel Modeling of Root Water Uptake. Proceedings 2010, JSC Guest Student Programme on Scientific Computing, 2010. link to proceedings