Exploratory Focus
Many physical, chemical or biological processes can be described by means of partial differential equations. If the underlying processes exhibit a nonlinear dynamic, an analysis and prediction of the complex behaviour is often only possible by solving the partial differential equations numerically. That is why the design of efficient numerical schemes is central to our research. In spite of the increasing computational capacities many problems are still only solvable with severe simplifications. This is due to the high complexity of the problems. One may think of e.g. the weather forecast and the huge amount of processes which have to be considered in this context. This example also makes clear, why it is of great importance to design efficient numerical schemes, i.e. that the numerical methods lead with least possible effort to the best possible solution: The weather forecast for yesterday is not very interesting at all.
Adaptive modelling and model reduction, adaptive grid refinement, multiscale methods and parallelisation are important methods to increase the efficiency of numerical schemes. But the quality of the numerical approximation is essential, too, meaning that we want to know how good the computed solution is in comparison to the exact one. In this context a posterior error estimates play a crucial role. They allow to estimate the approximation error without using the exact solution of the partial differential equation, which is generally unknown. In short we aim at developing numerical methods, which are as exact as possible and as fast as possible at the same time.
Furthermore, a posteriori error estimates can be used to design efficient schemes. One example is adaptive grid refinement. Here, we refine a computational grid only in regions where the local a posteriori error estimate is high. If the estimated error is small, the grid might even be coarsened. In this way, we can save computational costs and get more efficient numerical schemes.
In order to validate our numerical models, it is finally necessary to compare the numerical solution with measurements. For instance, if we want to check a model which forecasts the rainfall, we compare the predicted abundance of rain with the amount which has actually fallen. Thus, the constant feedback between numerical modelling and application is of particular importance. For this purpose we cooperate with partners from industry or clinical centres to check our modells for their applicability in actual problems. Within such projects we currently concentrate on fluid flow within technical applications, such as fuel cells, on hydrological processes for water management and on biomedical applications.
As aforementioned, also model reduction can be used to increase the efficiency of a numerical scheme. This can be done in different ways. On the one hand some processes can be neglected on purpose. For example we could disregard the effects of plants on the dispensation of water in the ground. On the other hand numerical methods can be simplified. An example is the Reduced Basis Method. It can be used for dealing with problems where certain parameters have to be changed often and fast, e.g. during the approach to land of a plane the approach angle of the landing airbrakes have to be adjusted. The idea of the Reduced Basis Method is to approximate the high dimensional solution space of conventional approximation methods like the Finite Element or the Finite Volume approach by a very low dimensional space, which however covers the whole spectrum of possible solutions with a sufficient accuracy. The generalization of the Reduced Basis Method on complex nonlinear evolution equations is vital to our research in this field.
Apart from the nonlinear dynamics of hydrological processes, the treatment of seperated scales constitutes another problem in dealing with the underlying differential equations. Concerning the spatial coordinate there exist processes with scales of kilometres (How does the water disperse in the ground?) and millimetres (How does the water pass a grain of sand?). Also in time there are such variations in scale, depending on the application. In order to deal with such processes, one can use multi scale methods, where the effects of the coarse (km) and the fine (mm) scales are coupled efficiently. The development of a posteriori error estimates for such multiscale methods is still in the early stages and central to our research.
In summary we can identify the following four major topics that form our exploratory focus:
A Posteriori Error Estimation and Adaptivity
Model Reduction with Reduced Basis Methods
Numerical Multiscale Algorithms
Model Reduction for Bayesian Inverse Problems
