Many modelling problems involve a very long time-horizont, e.g., the structural stability of a bridge that should last for many decades or that of a wind turbine that should last at least for a few decades. Both systems however undergo forcing on many different time-scales. By the wind, by temperature gradients by day and night, winter and summer, a wind turbine by the continuous rotation, the bridge by traffic, etc. Models for describing such systems are based on several PDEs and a long-term simulation that covers all scales (decades down to less than seconds) is not feasible. In recent time, Stefan Frei and Thomas Richter have developed a multiscale scheme for an efficient simulation of such problems based on averaged problems that involve short scale influences. This method fits into the framework of the heterogenous multiscale method and that allows for efficient simulations with significant speed ups. Fast and slow scales are decoupled by introducing local averages and by replacing fast scale contributions by localized periodic-in-time problems.

In our PhD project, we analyse such multiscale methods and in particular, we derive error estimators for an adaptive control of all discretization parameters (discretization in space, in time and control of the multiscale approximation). We consider systems of differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to further solution components that act on a fast scale. Although being an essential part of the coupled problem, these fast variables are often of no interest themselves.

For a first example we consider a system of ordinary differential equations. Here, we derive an a posteriori error estimator based on the dual weighted residual method that allows for a splitting of the error into averaging error, error of the slow scale and error of the fast scale. We demonstrate the accuracy of the error estimator and also its use for adaptive control of a multiscale scheme.

We then extend the theory to a system of differential equations with real life applications, for example plaque growth in blood vessels.