01.06.2024–31.05.2029
ERC project
Principal investigator: Michael FEISCHL (E101-02-3)

In order to describe complex phenomena in physics and technology, we often use partial differential equations which have to be solved with computer assistance. This has to be done very efficiently, otherwise relevant problems cannot be solved in a reasonable time. For example, it makes no sense to calculate the weather forecast for the next day if the calculation takes two days. Adaptive mesh refinement provides a remedy here by adapting the underlying approximation space (the finite element grid) with “a posteriori error estimators” to the problem.

In the ERC project “New Frontiers in Optimal Adaptivity”, we are now trying to show how this adaptation can be optimally achieved. The ideal algorithm thus provides the highest possible accuracy in a given computing time. This can be expressed in precice mathematical terms and there are very nice theoretical statements about the optimality of these algorithms. The problem is that this theory only works for stationary (i.e. not time-dependent) equations and therefore leaves out many practically important applications (such as the weather report). The time dependence inevitably disturbs many symmetries that are found in stationary equations and on which the classical proofs are based, and new ideas are needed for this.

Research Group: Computational PDEs (Fernando HENRIQUEZ, Amanda HUBER, David NIEDERKOFLER, Andrea SCAGLIONI, Fabian ZEHETGRUBER, Michael FEISCHL)