In recent years, a successful cooperation has developed between colleagues (in Germany and abroad) in the field of adaptive finite element methods to solve partial differential equations using error estimators. These methods allow the numerical simulation of complex, highly non-linear physical phenomena which would scarcely be possible without such strategies. Only by using mathematically justified algorithms of this type is it possible to utilise the computational resource of modern computers effectively.

As part of the collaboration, a posteriori error estimations for non-linear elliptical and (degenerate) parabolic problems and coupled systems were proven. In furtherance of this, adaptive finite element methods were implemented which document the practicability and efficiency of the results. Results have been published in the field of applications with degenerate parabolic problems with phase transitions and monotone operators. This also already touches on many aspects of the application projects listed below. The implementation of the methods is largely based on our ALBERTA toolbox, c.f. "ALBERTA - A Finite Element Toolbox for Research and Teaching".