Aaron Brunk

Analysis of PDEs, Numerical Analysis, Mathematical Modeling.

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I am a postdoctoral researcher at the Institute of Mathematics of the Johannes Gutenberg University in Mainz in the group of Mária Lukácová-Medvidová. My research is focused on the analysis and development of numerical methods for partial differential equations. In particular, I am interested in the stability of these schemes as well structure-preserving techniques, allowing to unify the analysis of the continuous and the discrete level.

Research interests:

  1. Numerical analysis
    • Numerical schemes for parabolic systems of partial differential equations: Variational methods, Continous Galerkin methods, Petrov-Galerkin methods
    • Structure-preserving methods: Conservation/dissipation of entropy/energy, variational reformulation, (skew-symmetric) approximation of convective terms
    • Error estimates: Nonlinear stability analysis, relative energy/entropy methods, a priori/posteriori analysis
    • Inverse Problems: Parameter identification and estimation
  2. Mathematical analysis
    • Wellposedness for parabolic systems of partial differential equation: Energy/Entropy methods; nonlinear and cross-diffusion systems
    • Nonlinear stability: relative energy/entropy methods; weak-strong uniqueness; Bregman divergence, Long-time behaviour of solutions
  3. Mathematical modelling
    • Thermodynamically consistent modelling: Variational systems; Onsager principles; macroscopic closure
    • Model reduction: Formal asymptotic limits;