Numerical Mathematics with Applications

Major research interests

Numerical methods for complex multiphysics and multiscale problems.

  • Adaptive Finite Elements. Adaptivity in finite element discretization is a flexible tool for reduction of the problem size. We study goal-oriented error estimators that can be applied to complex systems of equations and that are also transferred to error estimation of temporal discretization schemes;
  • Efficient Simulation Tools. We investigate parallel Newton-multigrid methods for accelerating the solution of algebraic systems arising from the discretization of complex partial differential equations. Furthermore, we consider multiscale-methods based on coupling continuum and particle systems;
  • Neural Networks in Physical Simulations. We explore the power of deep neural networks to augment classical simulation techniques. Often material properties cannot be exactly described by analytical principles, instead, measurement data enters the modeling. On the other hand, machine learning techniques are used to accelerate numerical algorithms, e.g. by learning preconditioners for linear solvers
  • Multiscale. Many problem feature a multiscale character, e.g. flow of a granulate material where the single particles are important for macroscopic behavior but too many to be resolved individually. Such a multiscale property is also given in time, like in mechanical fatigue processes, where fast small-scale influences can alter the long-term properties;
  • Bio-chemical and medical applications. Blood flow problems involve fluid-structure interactions but further biological and chemical processes as vessels might be damaged or infected and as growth, rupture or healing processes might take place. The resulting systems of partial differential equations are large and nonlinear such that efficient new methods are needed. Another application is the study of biological pattern formation processes as a self-stabilizing process in embryogenesis.

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