Name: Maximilian BRUNNER 
Current position: Postdoc (Project assistent, FWF) at ASC 
Work group: Numerics of PDEs (NumPDEs) (Prof. Dirk PRAETORIUS) 
Starting date: November 2020 as doctoral student, since May 2024 as Postdoc 
Topic of the dissertation: Über optimale Adaptivität für semilineare PDEs (defence in April 2024) 

The numerical solution of partial differential equations (PDEs) requires sophisticated numerical methods that weight up different sources of error and systematically reduce them. 

In my dissertation, I worked on adaptive finite element methods (FEM) for semilinear elliptic PDEs. The focus was on developing and analysing adaptive algorithms that ensure optimal convergence rates. Since semilinear problems contain nonlinear terms, the central challenge was to develop an algorithmic balance between linearisation and discretisation errors. 

This interplay of different error sources also represents a major challenge in my current research as a postdoctoral fellow. My work to date now enables me to investigate more complex issues that involve additional error sources, such as parabolic PDEs (temporal errors) or the consideration of uncertainties in modeling (parametric errors). The goal of my basic research is to enable reliable numerical solutions to increasingly realistic model problems. 

I joined Prof. Dirk Praetorius' work group because I particularly enjoyed the FEM lecture in my master's programme and this lies within his area of expertise. The opportunity to combine theoretical mathematics with practical applications appealed to me from the very beginning. I felt very comfortable in the NumPDEs group from the very start: the culture of feedback and the intensive exchange made my doctorate an extremely enriching experience. In addition, it also offered me valuable personal insights through international collaborations and conference visits, where I was able to present my own work and get to know the people behind the publications I had read. 

I also used my time as a postdoctoral researcher to write a research proposal aimed at a stay abroad. This should enable me to learn new ways of thinking and perspectives, prove myself outside my academic family, and further expand my scientific network. Since I enjoy both research and teaching, I am looking forward to the further development of my academic career.